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Integral of d{x}:
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Integral of x^2cosx^3
Identical expressions
x^2cosx^ three
x squared co sinus of e of x cubed
x squared co sinus of e of x to the power of three
x2cosx3
x²cosx³
x to the power of 2cosx to the power of 3
x^2cosx^3dx
Similar expressions
x^2*cos(x^3)*exp(x^3)
3*x^2*cos(x^3+5)

Solving integrals/ x^2cosx^3

Integral of x^2cosx^3 dx

The solution

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$$\int\limits_{0}^{1} x^{2} \cos^{3}{\left(x \right)}\, dx$$

Integral(x^2*cos(x)^3, (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \cos^{3}{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 2 x$ .

To find $v{\left(x \right)}$ :
1. Rewrite the integrand:
  
  $\cos^{3}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}$
2. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int \left(1 - u^{2}\right)\, du$
    1. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      The result is: $- \frac{u^{3}}{3} + u$
    Now substitute $u$ back in:
    
    $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
  Method #2
  1. Rewrite the integrand:
    
    $\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
      
      So, the result is: $- \frac{\sin^{3}{\left(x \right)}}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    The result is: $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
  Method #3
  1. Rewrite the integrand:
    
    $\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)} = - \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \sin{\left(x \right)}$ .
      
      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
      
      $\int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{3}{\left(x \right)}}{3}$
      
      So, the result is: $- \frac{\sin^{3}{\left(x \right)}}{3}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    The result is: $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \frac{2 x}{3}$ and let $\operatorname{dv}{\left(x \right)} = - \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = \frac{2}{3}$ .

To find $v{\left(x \right)}$ :
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \sin^{3}{\left(x \right)}\right)\, dx = - \int \sin^{3}{\left(x \right)}\, dx$
    1. Rewrite the integrand:
      
      $\sin^{3}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$
    2. Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $du$ :
      
      $\int \left(u^{2} - 1\right)\, du$
      1. Integrate term-by-term:
        
        The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u^{2}\, du = \frac{u^{3}}{3}$
        
        The integral of a constant is the constant times the variable of integration:
        
        $\int \left(-1\right)\, du = - u$
        
        The result is: $\frac{u^{3}}{3} - u$
      Now substitute $u$ back in:
      
      $\frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$
    So, the result is: $- \frac{\cos^{3}{\left(x \right)}}{3} + \cos{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \sin{\left(x \right)}\, dx = 3 \int \sin{\left(x \right)}\, dx$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    So, the result is: $- 3 \cos{\left(x \right)}$
  The result is: $- \frac{\cos^{3}{\left(x \right)}}{3} - 2 \cos{\left(x \right)}$
Now evaluate the sub-integral.
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{2 \cos^{3}{\left(x \right)}}{9}\right)\, dx = - \frac{2 \int \cos^{3}{\left(x \right)}\, dx}{9}$
  1. Rewrite the integrand:
    
    $\cos^{3}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}$
  2. Let $u = \sin{\left(x \right)}$ .
    
    Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
    
    $\int \left(1 - u^{2}\right)\, du$
    1. Integrate term-by-term:
      1. The integral of a constant is the constant times the variable of integration:
        
        $\int 1\, du = u$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
        
        The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u^{2}\, du = \frac{u^{3}}{3}$
        
        So, the result is: $- \frac{u^{3}}{3}$
      The result is: $- \frac{u^{3}}{3} + u$
    Now substitute $u$ back in:
    
    $- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}$
  So, the result is: $\frac{2 \sin^{3}{\left(x \right)}}{27} - \frac{2 \sin{\left(x \right)}}{9}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{4 \cos{\left(x \right)}}{3}\right)\, dx = - \frac{4 \int \cos{\left(x \right)}\, dx}{3}$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $- \frac{4 \sin{\left(x \right)}}{3}$
The result is: $\frac{2 \sin^{3}{\left(x \right)}}{27} - \frac{14 \sin{\left(x \right)}}{9}$
Now simplify:

$\frac{x^{2} \left(\cos^{2}{\left(x \right)} + 2\right) \sin{\left(x \right)}}{3} + \frac{2 x \left(\cos^{2}{\left(x \right)} + 6\right) \cos{\left(x \right)}}{9} + \frac{2 \sin^{3}{\left(x \right)}}{27} - \frac{14 \sin{\left(x \right)}}{9}$
Add the constant of integration:

$\frac{x^{2} \left(\cos^{2}{\left(x \right)} + 2\right) \sin{\left(x \right)}}{3} + \frac{2 x \left(\cos^{2}{\left(x \right)} + 6\right) \cos{\left(x \right)}}{9} + \frac{2 \sin^{3}{\left(x \right)}}{27} - \frac{14 \sin{\left(x \right)}}{9}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(\cos^{2}{\left(x \right)} + 2\right) \sin{\left(x \right)}}{3} + \frac{2 x \left(\cos^{2}{\left(x \right)} + 6\right) \cos{\left(x \right)}}{9} + \frac{2 \sin^{3}{\left(x \right)}}{27} - \frac{14 \sin{\left(x \right)}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                                                             /               3   \
  /                                                                          |            cos (x)|
 |                                      3         /     3            \   2*x*|-2*cos(x) - -------|
 |  2    3             14*sin(x)   2*sin (x)    2 |  sin (x)         |       \               3   /
 | x *cos (x) dx = C - --------- + --------- + x *|- ------- + sin(x)| - -------------------------
 |                         9           27         \     3            /               3            
/

$${{\left(9\,x^2-2\right)\,\sin \left(3\,x\right)+6\,x\,\cos \left(3 \,x\right)+\left(81\,x^2-162\right)\,\sin x+162\,x\,\cos x}\over{108 }}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        3            3           2                  2          
  22*sin (1)   14*cos (1)   5*cos (1)*sin(1)   4*sin (1)*cos(1)
- ---------- + ---------- - ---------------- + ----------------
      27           9               9                  3

$${{7\,\sin 3+6\,\cos 3-81\,\sin 1+162\,\cos 1}\over{108}}$$

        3            3           2                  2          
  22*sin (1)   14*cos (1)   5*cos (1)*sin(1)   4*sin (1)*cos(1)
- ---------- + ---------- - ---------------- + ----------------
      27           9               9                  3

$$- \frac{22 \sin^{3}{\left(1 \right)}}{27} - \frac{5 \sin{\left(1 \right)} \cos^{2}{\left(1 \right)}}{9} + \frac{14 \cos^{3}{\left(1 \right)}}{9} + \frac{4 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{3}$$

Simplify

Numerical answer [src]

0.133497304240883

0.133497304240883

The graph

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

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How to use it?

Identical expressions

Similar expressions

Integral of x^2cosx^3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3