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Derivative of:
x^5*cos(x)
Limit of the function:
x^5*cos(x)
Identical expressions
x^ five *cos(x)
x to the power of 5 multiply by co sinus of e of (x)
x to the power of five multiply by co sinus of e of (x)
x5*cos(x)
x5*cosx
x⁵*cos(x)
x^5cos(x)
x5cos(x)
x5cosx
x^5cosx
x^5*cos(x)dx
Similar expressions
x^5*cosx

Solving integrals/ x^5*cos(x)

Integral of x^5*cos(x) dx

The solution

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  1             
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 |   5          
 |  x *cos(x) dx
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0

\int\limits_{0}^{1} x^{5} \cos{\left(x \right)}\, dx

Integral(x^5*cos(x), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = x^{5}$ 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]

-120 + 65*cos(1) + 101*sin(1)

-120 + 65 \cos{\left(1 \right)} + 101 \sin{\left(1 \right)}

-120 + 65*cos(1) + 101*sin(1)

-120 + 65 \cos{\left(1 \right)} + 101 \sin{\left(1 \right)}

-120 + 65*cos(1) + 101*sin(1)

Numerical answer [src]

0.108219347026629

0.108219347026629

The graph

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

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Identical expressions

Similar expressions

Integral of x^5*cos(x) dx

Limits of integration:

The graph:

Piecewise:

The solution