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Integral of d{x}:
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Derivative of:
x^5cosx
Identical expressions
x^5cosx
x to the power of 5 co sinus of e of x
x5cosx
x⁵cosx
x^5cosxdx

Integral of x^5cosx dx

The solution

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 pi             
 --             
 3              
  /             
 |              
 |   5          
 |  x *cos(x) dx
 |              
/               
0

$$\int\limits_{0}^{\frac{\pi}{3}} x^{5} \cos{\left(x \right)}\, dx$$

Integral(x^5*cos(x), (x, 0, pi/3))

Detail solution

Use integration by parts:

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

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

Then $\operatorname{du}{\left(x \right)} = 5 x^{4}$ .

To find $v{\left(x \right)}$ :
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = 5 x^{4}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 20 x^{3}$ .

To find $v{\left(x \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = - 20 x^{3}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - 60 x^{2}$ .

To find $v{\left(x \right)}$ :
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = - 60 x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - 120 x$ .

To find $v{\left(x \right)}$ :
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = 120 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 120$ .

To find $v{\left(x \right)}$ :
1. The integral of cosine is sine:
  
  $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int 120 \sin{\left(x \right)}\, dx = 120 \int \sin{\left(x \right)}\, dx$
1. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
So, the result is: $- 120 \cos{\left(x \right)}$
Add the constant of integration:

$x^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)} - 20 x^{3} \sin{\left(x \right)} - 60 x^{2} \cos{\left(x \right)} + 120 x \sin{\left(x \right)} + 120 \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$x^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)} - 20 x^{3} \sin{\left(x \right)} - 60 x^{2} \cos{\left(x \right)} + 120 x \sin{\left(x \right)} + 120 \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                    
 |                                                                                                     
 |  5                               5              2              3             4                      
 | x *cos(x) dx = C + 120*cos(x) + x *sin(x) - 60*x *cos(x) - 20*x *sin(x) + 5*x *cos(x) + 120*x*sin(x)
 |                                                                                                     
/

$$\int x^{5} \cos{\left(x \right)}\, dx = C + x^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)} - 20 x^{3} \sin{\left(x \right)} - 60 x^{2} \cos{\left(x \right)} + 120 x \sin{\left(x \right)} + 120 \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

           2       4                      ___   3     ___   5
      10*pi    5*pi            ___   10*\/ 3 *pi    \/ 3 *pi 
-60 - ------ + ----- + 20*pi*\/ 3  - ------------ + ---------
        3       162                       27           486

$$-60 - \frac{10 \pi^{2}}{3} - \frac{10 \sqrt{3} \pi^{3}}{27} + \frac{\sqrt{3} \pi^{5}}{486} + \frac{5 \pi^{4}}{162} + 20 \sqrt{3} \pi$$

           2       4                      ___   3     ___   5
      10*pi    5*pi            ___   10*\/ 3 *pi    \/ 3 *pi 
-60 - ------ + ----- + 20*pi*\/ 3  - ------------ + ---------
        3       162                       27           486

$$-60 - \frac{10 \pi^{2}}{3} - \frac{10 \sqrt{3} \pi^{3}}{27} + \frac{\sqrt{3} \pi^{5}}{486} + \frac{5 \pi^{4}}{162} + 20 \sqrt{3} \pi$$

-60 - 10*pi^2/3 + 5*pi^4/162 + 20*pi*sqrt(3) - 10*sqrt(3)*pi^3/27 + sqrt(3)*pi^5/486

Numerical answer [src]

0.135818842576677

0.135818842576677

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

Other calculators

How to use it?

Identical expressions

Integral of x^5cosx dx

Limits of integration:

The graph:

Piecewise:

The solution