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Identical expressions
sin(zero .1x)+x^ two +(eleven +x+(x^ three)/ three -1 zero cos(0.1x))^ two
sinus of (0.1x) plus x squared plus (11 plus x plus (x cubed ) divide by 3 minus 10 co sinus of e of (0.1x)) squared
sinus of (zero .1x) plus x to the power of two plus (eleven plus x plus (x to the power of three) divide by three minus 1 zero co sinus of e of (0.1x)) to the power of two
sin(0.1x)+x2+(11+x+(x3)/3-10cos(0.1x))2
sin0.1x+x2+11+x+x3/3-10cos0.1x2
sin(0.1x)+x²+(11+x+(x³)/3-10cos(0.1x))²
sin(0.1x)+x to the power of 2+(11+x+(x to the power of 3)/3-10cos(0.1x)) to the power of 2
sin0.1x+x^2+11+x+x^3/3-10cos0.1x^2
sin(0.1x)+x^2+(11+x+(x^3) divide by 3-10cos(0.1x))^2
sin(0.1x)+x^2+(11+x+(x^3)/3-10cos(0.1x))^2dx
Similar expressions
sin(0.1x)+x^2+(11+x+(x^3)/3+10cos(0.1x))^2
sin(0.1x)+x^2+(11-x+(x^3)/3-10cos(0.1x))^2
sin(0.1x)+x^2+(11+x-(x^3)/3-10cos(0.1x))^2
sin(0.1x)-x^2+(11+x+(x^3)/3-10cos(0.1x))^2
sin(0.1x)+x^2-(11+x+(x^3)/3-10cos(0.1x))^2

Solving integrals/ sin(0.1x)+x^2+(11+x+(x^3)/3-10cos(0.1x))^2

Integral of sin(0.1x)+x^2+(11+x+(x^3)/3-10cos(0.1x))^2 dx

The solution

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$$\int\limits_{0}^{\frac{1}{10}} \left(\left(x^{2} + \sin{\left(\frac{x}{10} \right)}\right) + \left(\left(\frac{x^{3}}{3} + \left(x + 11\right)\right) - 10 \cos{\left(\frac{x}{10} \right)}\right)^{2}\right)\, dx$$

Integral(sin(x/10) + x^2 + (11 + x + x^3/3 - 10*cos(x/10))^2, (x, 0, 1/10))

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  1. Let $u = \frac{x}{10}$ .
    
    Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
    
    $\int 10 \sin{\left(u \right)}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 10 \int \sin{\left(u \right)}\, du$
      1. The integral of sine is negative cosine:
        
        $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      So, the result is: $- 10 \cos{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- 10 \cos{\left(\frac{x}{10} \right)}$
  The result is: $\frac{x^{3}}{3} - 10 \cos{\left(\frac{x}{10} \right)}$
1. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $\left(\left(\frac{x^{3}}{3} + \left(x + 11\right)\right) - 10 \cos{\left(\frac{x}{10} \right)}\right)^{2} = \frac{x^{6}}{9} + \frac{2 x^{4}}{3} - \frac{20 x^{3} \cos{\left(\frac{x}{10} \right)}}{3} + \frac{22 x^{3}}{3} + x^{2} - 20 x \cos{\left(\frac{x}{10} \right)} + 22 x + 100 \cos^{2}{\left(\frac{x}{10} \right)} - 220 \cos{\left(\frac{x}{10} \right)} + 121$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x^{6}}{9}\, dx = \frac{\int x^{6}\, dx}{9}$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{6}\, dx = \frac{x^{7}}{7}$
      
      So, the result is: $\frac{x^{7}}{63}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2 x^{4}}{3}\, dx = \frac{2 \int x^{4}\, dx}{3}$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{4}\, dx = \frac{x^{5}}{5}$
      
      So, the result is: $\frac{2 x^{5}}{15}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{20 x^{3} \cos{\left(\frac{x}{10} \right)}}{3}\right)\, dx = - \frac{20 \int x^{3} \cos{\left(\frac{x}{10} \right)}\, dx}{3}$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x^{3}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{10} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 3 x^{2}$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 10 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $10 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $10 \sin{\left(\frac{x}{10} \right)}$
      
      Now evaluate the sub-integral.
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = 30 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(\frac{x}{10} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 60 x$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \sin{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 10 \int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- 10 \cos{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- 10 \cos{\left(\frac{x}{10} \right)}$
      
      Now evaluate the sub-integral.
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = - 600 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{10} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = -600$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 10 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $10 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $10 \sin{\left(\frac{x}{10} \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 6000 \sin{\left(\frac{x}{10} \right)}\right)\, dx = - 6000 \int \sin{\left(\frac{x}{10} \right)}\, dx$
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \sin{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 10 \int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- 10 \cos{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- 10 \cos{\left(\frac{x}{10} \right)}$
      
      So, the result is: $60000 \cos{\left(\frac{x}{10} \right)}$
      
      So, the result is: $- \frac{200 x^{3} \sin{\left(\frac{x}{10} \right)}}{3} - 2000 x^{2} \cos{\left(\frac{x}{10} \right)} + 40000 x \sin{\left(\frac{x}{10} \right)} + 400000 \cos{\left(\frac{x}{10} \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{22 x^{3}}{3}\, dx = \frac{22 \int x^{3}\, dx}{3}$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{3}\, dx = \frac{x^{4}}{4}$
      
      So, the result is: $\frac{11 x^{4}}{6}$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 20 x \cos{\left(\frac{x}{10} \right)}\right)\, dx = - 20 \int x \cos{\left(\frac{x}{10} \right)}\, dx$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{10} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 10 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $10 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $10 \sin{\left(\frac{x}{10} \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 10 \sin{\left(\frac{x}{10} \right)}\, dx = 10 \int \sin{\left(\frac{x}{10} \right)}\, dx$
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \sin{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 10 \int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- 10 \cos{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- 10 \cos{\left(\frac{x}{10} \right)}$
      
      So, the result is: $- 100 \cos{\left(\frac{x}{10} \right)}$
      
      So, the result is: $- 200 x \sin{\left(\frac{x}{10} \right)} - 2000 \cos{\left(\frac{x}{10} \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 22 x\, dx = 22 \int x\, dx$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      So, the result is: $11 x^{2}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 100 \cos^{2}{\left(\frac{x}{10} \right)}\, dx = 100 \int \cos^{2}{\left(\frac{x}{10} \right)}\, dx$
      
      Rewrite the integrand:
      
      $\cos^{2}{\left(\frac{x}{10} \right)} = \frac{\cos{\left(\frac{x}{5} \right)}}{2} + \frac{1}{2}$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(\frac{x}{5} \right)}}{2}\, dx = \frac{\int \cos{\left(\frac{x}{5} \right)}\, dx}{2}$
      
      Let $u = \frac{x}{5}$ .
      
      Then let $du = \frac{dx}{5}$ and substitute $5 du$ :
      
      $\int 5 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 5 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $5 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $5 \sin{\left(\frac{x}{5} \right)}$
      
      So, the result is: $\frac{5 \sin{\left(\frac{x}{5} \right)}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{5 \sin{\left(\frac{x}{5} \right)}}{2}$
      
      So, the result is: $50 x + 250 \sin{\left(\frac{x}{5} \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 220 \cos{\left(\frac{x}{10} \right)}\right)\, dx = - 220 \int \cos{\left(\frac{x}{10} \right)}\, dx$
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 10 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $10 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $10 \sin{\left(\frac{x}{10} \right)}$
      
      So, the result is: $- 2200 \sin{\left(\frac{x}{10} \right)}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 121\, dx = 121 x$
    The result is: $\frac{x^{7}}{63} + \frac{2 x^{5}}{15} + \frac{11 x^{4}}{6} - \frac{200 x^{3} \sin{\left(\frac{x}{10} \right)}}{3} + \frac{x^{3}}{3} - 2000 x^{2} \cos{\left(\frac{x}{10} \right)} + 11 x^{2} + 39800 x \sin{\left(\frac{x}{10} \right)} + 171 x - 2200 \sin{\left(\frac{x}{10} \right)} + 250 \sin{\left(\frac{x}{5} \right)} + 398000 \cos{\left(\frac{x}{10} \right)}$
  Method #2
  1. Rewrite the integrand:
    
    $\left(\left(\frac{x^{3}}{3} + \left(x + 11\right)\right) - 10 \cos{\left(\frac{x}{10} \right)}\right)^{2} = \frac{x^{6}}{9} + \frac{2 x^{4}}{3} - \frac{20 x^{3} \cos{\left(\frac{x}{10} \right)}}{3} + \frac{22 x^{3}}{3} + x^{2} - 20 x \cos{\left(\frac{x}{10} \right)} + 22 x + 100 \cos^{2}{\left(\frac{x}{10} \right)} - 220 \cos{\left(\frac{x}{10} \right)} + 121$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{x^{6}}{9}\, dx = \frac{\int x^{6}\, dx}{9}$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{6}\, dx = \frac{x^{7}}{7}$
      
      So, the result is: $\frac{x^{7}}{63}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2 x^{4}}{3}\, dx = \frac{2 \int x^{4}\, dx}{3}$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{4}\, dx = \frac{x^{5}}{5}$
      
      So, the result is: $\frac{2 x^{5}}{15}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{20 x^{3} \cos{\left(\frac{x}{10} \right)}}{3}\right)\, dx = - \frac{20 \int x^{3} \cos{\left(\frac{x}{10} \right)}\, dx}{3}$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x^{3}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{10} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 3 x^{2}$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 10 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $10 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $10 \sin{\left(\frac{x}{10} \right)}$
      
      Now evaluate the sub-integral.
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = 30 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(\frac{x}{10} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 60 x$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \sin{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 10 \int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- 10 \cos{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- 10 \cos{\left(\frac{x}{10} \right)}$
      
      Now evaluate the sub-integral.
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = - 600 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{10} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = -600$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 10 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $10 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $10 \sin{\left(\frac{x}{10} \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 6000 \sin{\left(\frac{x}{10} \right)}\right)\, dx = - 6000 \int \sin{\left(\frac{x}{10} \right)}\, dx$
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \sin{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 10 \int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- 10 \cos{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- 10 \cos{\left(\frac{x}{10} \right)}$
      
      So, the result is: $60000 \cos{\left(\frac{x}{10} \right)}$
      
      So, the result is: $- \frac{200 x^{3} \sin{\left(\frac{x}{10} \right)}}{3} - 2000 x^{2} \cos{\left(\frac{x}{10} \right)} + 40000 x \sin{\left(\frac{x}{10} \right)} + 400000 \cos{\left(\frac{x}{10} \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{22 x^{3}}{3}\, dx = \frac{22 \int x^{3}\, dx}{3}$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{3}\, dx = \frac{x^{4}}{4}$
      
      So, the result is: $\frac{11 x^{4}}{6}$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 20 x \cos{\left(\frac{x}{10} \right)}\right)\, dx = - 20 \int x \cos{\left(\frac{x}{10} \right)}\, dx$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{10} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 10 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $10 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $10 \sin{\left(\frac{x}{10} \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 10 \sin{\left(\frac{x}{10} \right)}\, dx = 10 \int \sin{\left(\frac{x}{10} \right)}\, dx$
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \sin{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 10 \int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- 10 \cos{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- 10 \cos{\left(\frac{x}{10} \right)}$
      
      So, the result is: $- 100 \cos{\left(\frac{x}{10} \right)}$
      
      So, the result is: $- 200 x \sin{\left(\frac{x}{10} \right)} - 2000 \cos{\left(\frac{x}{10} \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 22 x\, dx = 22 \int x\, dx$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      So, the result is: $11 x^{2}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 100 \cos^{2}{\left(\frac{x}{10} \right)}\, dx = 100 \int \cos^{2}{\left(\frac{x}{10} \right)}\, dx$
      
      Rewrite the integrand:
      
      $\cos^{2}{\left(\frac{x}{10} \right)} = \frac{\cos{\left(\frac{x}{5} \right)}}{2} + \frac{1}{2}$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(\frac{x}{5} \right)}}{2}\, dx = \frac{\int \cos{\left(\frac{x}{5} \right)}\, dx}{2}$
      
      Let $u = \frac{x}{5}$ .
      
      Then let $du = \frac{dx}{5}$ and substitute $5 du$ :
      
      $\int 5 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 5 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $5 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $5 \sin{\left(\frac{x}{5} \right)}$
      
      So, the result is: $\frac{5 \sin{\left(\frac{x}{5} \right)}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{5 \sin{\left(\frac{x}{5} \right)}}{2}$
      
      So, the result is: $50 x + 250 \sin{\left(\frac{x}{5} \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 220 \cos{\left(\frac{x}{10} \right)}\right)\, dx = - 220 \int \cos{\left(\frac{x}{10} \right)}\, dx$
      
      Let $u = \frac{x}{10}$ .
      
      Then let $du = \frac{dx}{10}$ and substitute $10 du$ :
      
      $\int 10 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = 10 \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $10 \sin{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $10 \sin{\left(\frac{x}{10} \right)}$
      
      So, the result is: $- 2200 \sin{\left(\frac{x}{10} \right)}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 121\, dx = 121 x$
    The result is: $\frac{x^{7}}{63} + \frac{2 x^{5}}{15} + \frac{11 x^{4}}{6} - \frac{200 x^{3} \sin{\left(\frac{x}{10} \right)}}{3} + \frac{x^{3}}{3} - 2000 x^{2} \cos{\left(\frac{x}{10} \right)} + 11 x^{2} + 39800 x \sin{\left(\frac{x}{10} \right)} + 171 x - 2200 \sin{\left(\frac{x}{10} \right)} + 250 \sin{\left(\frac{x}{5} \right)} + 398000 \cos{\left(\frac{x}{10} \right)}$
The result is: $\frac{x^{7}}{63} + \frac{2 x^{5}}{15} + \frac{11 x^{4}}{6} - \frac{200 x^{3} \sin{\left(\frac{x}{10} \right)}}{3} + \frac{2 x^{3}}{3} - 2000 x^{2} \cos{\left(\frac{x}{10} \right)} + 11 x^{2} + 39800 x \sin{\left(\frac{x}{10} \right)} + 171 x - 2200 \sin{\left(\frac{x}{10} \right)} + 250 \sin{\left(\frac{x}{5} \right)} + 397990 \cos{\left(\frac{x}{10} \right)}$
Add the constant of integration:

$\frac{x^{7}}{63} + \frac{2 x^{5}}{15} + \frac{11 x^{4}}{6} - \frac{200 x^{3} \sin{\left(\frac{x}{10} \right)}}{3} + \frac{2 x^{3}}{3} - 2000 x^{2} \cos{\left(\frac{x}{10} \right)} + 11 x^{2} + 39800 x \sin{\left(\frac{x}{10} \right)} + 171 x - 2200 \sin{\left(\frac{x}{10} \right)} + 250 \sin{\left(\frac{x}{5} \right)} + 397990 \cos{\left(\frac{x}{10} \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{7}}{63} + \frac{2 x^{5}}{15} + \frac{11 x^{4}}{6} - \frac{200 x^{3} \sin{\left(\frac{x}{10} \right)}}{3} + \frac{2 x^{3}}{3} - 2000 x^{2} \cos{\left(\frac{x}{10} \right)} + 11 x^{2} + 39800 x \sin{\left(\frac{x}{10} \right)} + 171 x - 2200 \sin{\left(\frac{x}{10} \right)} + 250 \sin{\left(\frac{x}{5} \right)} + 397990 \cos{\left(\frac{x}{10} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                                                                                
 |                                                                                                                                                                                                 
 | /                                         2\                                                                                                                                           3    /x \
 | |               /          3             \ |                                                                        7      3      5       4                                       200*x *sin|--|
 | |   /x \    2   |         x          /x \| |                  /x \       2                  /x\             /x \   x    2*x    2*x    11*x          2    /x \              /x \             \10/
 | |sin|--| + x  + |11 + x + -- - 10*cos|--|| | dx = C - 2200*sin|--| + 11*x  + 171*x + 250*sin|-| + 397990*cos|--| + -- + ---- + ---- + ----- - 2000*x *cos|--| + 39800*x*sin|--| - --------------
 | \   \10/        \         3          \10// /                  \10/                          \5/             \10/   63    3      15      6                \10/              \10/         3       
 |                                                                                                                                                                                                 
/

$$\int \left(\left(x^{2} + \sin{\left(\frac{x}{10} \right)}\right) + \left(\left(\frac{x^{3}}{3} + \left(x + 11\right)\right) - 10 \cos{\left(\frac{x}{10} \right)}\right)^{2}\right)\, dx = C + \frac{x^{7}}{63} + \frac{2 x^{5}}{15} + \frac{11 x^{4}}{6} - \frac{200 x^{3} \sin{\left(\frac{x}{10} \right)}}{3} + \frac{2 x^{3}}{3} - 2000 x^{2} \cos{\left(\frac{x}{10} \right)} + 11 x^{2} + 39800 x \sin{\left(\frac{x}{10} \right)} + 171 x - 2200 \sin{\left(\frac{x}{10} \right)} + 250 \sin{\left(\frac{x}{5} \right)} + 397990 \cos{\left(\frac{x}{10} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  250726007163659        2               2                              26699*sin(1/100)                            
- --------------- + 5*cos (1/100) + 5*sin (1/100) + 397970*cos(1/100) + ---------------- + 500*cos(1/100)*sin(1/100)
     630000000                                                                 15

$$- \frac{250726007163659}{630000000} + 5 \sin^{2}{\left(\frac{1}{100} \right)} + 5 \cos^{2}{\left(\frac{1}{100} \right)} + 500 \sin{\left(\frac{1}{100} \right)} \cos{\left(\frac{1}{100} \right)} + \frac{26699 \sin{\left(\frac{1}{100} \right)}}{15} + 397970 \cos{\left(\frac{1}{100} \right)}$$

  250726007163659        2               2                              26699*sin(1/100)                            
- --------------- + 5*cos (1/100) + 5*sin (1/100) + 397970*cos(1/100) + ---------------- + 500*cos(1/100)*sin(1/100)
     630000000                                                                 15

-250726007163659/630000000 + 5*cos(1/100)^2 + 5*sin(1/100)^2 + 397970*cos(1/100) + 26699*sin(1/100)/15 + 500*cos(1/100)*sin(1/100)

Numerical answer [src]

0.111220507795555

0.111220507795555

The graph

Integral of sin(0.1x)+x^2+(11+x+(x^3)/3-10cos(0.1x))^2 dx

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sin(0.1x)+x^2+(11+x+(x^3)/3-10cos(0.1x))^2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2