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Solving integrals/ x^2sinx^3dx

Integral of x^2sinx^3dx dx

The solution

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 |  x *sin (x)*1 dx
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0

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

Integral(x^2*sin(x)^3*1, (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 $\operatorname{dv}{\left(x \right)} = \sin^{3}{\left(x \right)}$ .

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

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

                                                                              /               3   \
  /                                                                           |            sin (x)|
 |                            3                     /             3   \   2*x*|-2*sin(x) - -------|
 |  2    3               2*cos (x)   14*cos(x)    2 |          cos (x)|       \               3   /
 | x *sin (x)*1 dx = C - --------- + --------- + x *|-cos(x) + -------| - -------------------------
 |                           27          9          \             3   /               3            
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

             3            3           2                  2          
  40   14*sin (1)   22*cos (1)   4*cos (1)*sin(1)   5*sin (1)*cos(1)
- -- + ---------- + ---------- + ---------------- + ----------------
  27       9            27              3                  9

$$-{{6\,\sin 3-7\,\cos 3-162\,\sin 1-81\,\cos 1}\over{108}}-{{40 }\over{27}}$$

             3            3           2                  2          
  40   14*sin (1)   22*cos (1)   4*cos (1)*sin(1)   5*sin (1)*cos(1)
- -- + ---------- + ---------- + ---------------- + ----------------
  27       9            27              3                  9

$$- \frac{40}{27} + \frac{22 \cos^{3}{\left(1 \right)}}{27} + \frac{5 \sin^{2}{\left(1 \right)} \cos{\left(1 \right)}}{9} + \frac{4 \sin{\left(1 \right)} \cos^{2}{\left(1 \right)}}{3} + \frac{14 \sin^{3}{\left(1 \right)}}{9}$$

Simplify

Numerical answer [src]

0.113945544348484

0.113945544348484

The graph

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

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Identical expressions

Similar expressions

Integral of x^2sinx^3dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3