Integral of x^2sin4x dx

The solution

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  0               
  /               
 |                
 |   2            
 |  x *sin(4*x) dx
 |                
/                 
pi                
--                
8

$$\int\limits_{\frac{\pi}{8}}^{0} x^{2} \sin{\left(4 x \right)}\, dx$$

Integral(x^2*sin(4*x), (x, pi/8, 0))

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 $\operatorname{dv}{\left(x \right)} = \sin{\left(4 x \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{\sin{\left(u \right)}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(4 x \right)}}{4}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = - \frac{x}{2}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(4 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - \frac{1}{2}$ .

To find $v{\left(x \right)}$ :
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{\cos{\left(u \right)}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{4}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(4 x \right)}}{4}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{\sin{\left(4 x \right)}}{8}\right)\, dx = - \frac{\int \sin{\left(4 x \right)}\, dx}{8}$
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{\sin{\left(u \right)}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{4}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(4 x \right)}}{4}$
So, the result is: $\frac{\cos{\left(4 x \right)}}{32}$
Add the constant of integration:

$- \frac{x^{2} \cos{\left(4 x \right)}}{4} + \frac{x \sin{\left(4 x \right)}}{8} + \frac{\cos{\left(4 x \right)}}{32}+ \mathrm{constant}$

The answer is:

$- \frac{x^{2} \cos{\left(4 x \right)}}{4} + \frac{x \sin{\left(4 x \right)}}{8} + \frac{\cos{\left(4 x \right)}}{32}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                        
 |                                  2                      
 |  2                   cos(4*x)   x *cos(4*x)   x*sin(4*x)
 | x *sin(4*x) dx = C + -------- - ----------- + ----------
 |                         32           4            8     
/

$$\int x^{2} \sin{\left(4 x \right)}\, dx = C - \frac{x^{2} \cos{\left(4 x \right)}}{4} + \frac{x \sin{\left(4 x \right)}}{8} + \frac{\cos{\left(4 x \right)}}{32}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1    pi
-- - --
32   64

$$\frac{1}{32} - \frac{\pi}{64}$$

1    pi
-- - --
32   64

$$\frac{1}{32} - \frac{\pi}{64}$$

1/32 - pi/64

Numerical answer [src]

-0.0178373852123405

-0.0178373852123405

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

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Integral of x^2sin4x dx

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The solution