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Identical expressions
x^ two *cos(4x)
x squared multiply by co sinus of e of (4x)
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x2*cos(4x)
x2*cos4x
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Similar expressions
x^2*cos4(x)

Integral of x^2*cos(4x) dx

The solution

You have entered [src]

 2*p              
  /               
 |                
 |   2            
 |  x *cos(4*x) dx
 |                
/                 
0

$$\int\limits_{0}^{2 p} x^{2} \cos{\left(4 x \right)}\, dx$$

Integral(x^2*cos(4*x), (x, 0, 2*p))

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

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 $\frac{du}{4}$ :
  
  $\int \frac{\cos{\left(u \right)}}{16}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(u \right)}}{4}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
    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.
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 $\operatorname{dv}{\left(x \right)} = \sin{\left(4 x \right)}$ .

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 $\frac{du}{4}$ :
  
  $\int \frac{\sin{\left(u \right)}}{16}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{4}\, du = \frac{\int \sin{\left(u \right)}\, du}{4}$
    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.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{\cos{\left(4 x \right)}}{8}\right)\, dx = - \frac{\int \cos{\left(4 x \right)}\, dx}{8}$
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{\cos{\left(u \right)}}{16}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(u \right)}}{4}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
    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:
  
  $\frac{\sin{\left(4 x \right)}}{4}$
So, the result is: $- \frac{\sin{\left(4 x \right)}}{32}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

  sin(8*p)    2            p*cos(8*p)
- -------- + p *sin(8*p) + ----------
     32                        4

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

  sin(8*p)    2            p*cos(8*p)
- -------- + p *sin(8*p) + ----------
     32                        4

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

Упростить

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

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Integral of x^2*cos(4x) dx

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The solution