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Integral of sin^5
Derivative of:
x^3*cos(5x)
Identical expressions
x^ three *cos(5x)
x cubed multiply by co sinus of e of (5x)
x to the power of three multiply by co sinus of e of (5x)
x3*cos(5x)
x3*cos5x
x³*cos(5x)
x to the power of 3*cos(5x)
x^3cos(5x)
x3cos(5x)
x3cos5x
x^3cos5x
x^3*cos(5x)dx

Integral of x^3*cos(5x) dx

The solution

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 pi               
  /               
 |                
 |   3            
 |  x *cos(5*x) dx
 |                
/                 
-pi

\int\limits_{- \pi}^{\pi} x^{3} \cos{\left(5 x \right)}\, dx

Integral(x^3*cos(5*x), (x, -pi, pi))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = x^{3}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(5 x \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\cos{\left(u \right)}}{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(5 x \right)}}{5}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = \frac{3 x^{2}}{5}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(5 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = \frac{6 x}{5}$ .

To find $v{\left(x \right)}$ :
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\sin{\left(u \right)}}{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(5 x \right)}}{5}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = - \frac{6 x}{25}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(5 x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - \frac{6}{25}$ .

To find $v{\left(x \right)}$ :
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\cos{\left(u \right)}}{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(5 x \right)}}{5}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{6 \sin{\left(5 x \right)}}{125}\right)\, dx = - \frac{6 \int \sin{\left(5 x \right)}\, dx}{125}$
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\sin{\left(u \right)}}{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{5}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(5 x \right)}}{5}$
So, the result is: $\frac{6 \cos{\left(5 x \right)}}{625}$
Add the constant of integration:

$\frac{x^{3} \sin{\left(5 x \right)}}{5} + \frac{3 x^{2} \cos{\left(5 x \right)}}{25} - \frac{6 x \sin{\left(5 x \right)}}{125} - \frac{6 \cos{\left(5 x \right)}}{625}+ \mathrm{constant}$

The answer is:

\frac{x^{3} \sin{\left(5 x \right)}}{5} + \frac{3 x^{2} \cos{\left(5 x \right)}}{25} - \frac{6 x \sin{\left(5 x \right)}}{125} - \frac{6 \cos{\left(5 x \right)}}{625}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                                                            
 |                                                   3               2         
 |  3                   6*cos(5*x)   6*x*sin(5*x)   x *sin(5*x)   3*x *cos(5*x)
 | x *cos(5*x) dx = C - ---------- - ------------ + ----------- + -------------
 |                         625           125             5              25     
/

\int x^{3} \cos{\left(5 x \right)}\, dx = C + \frac{x^{3} \sin{\left(5 x \right)}}{5} + \frac{3 x^{2} \cos{\left(5 x \right)}}{25} - \frac{6 x \sin{\left(5 x \right)}}{125} - \frac{6 \cos{\left(5 x \right)}}{625}

Simplify

The graph

Plot the graph f(x)

The answer [src]

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Numerical answer [src]

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

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Identical expressions

Integral of x^3*cos(5x) dx

Limits of integration:

The graph:

Piecewise:

The solution