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x^5*cos(x)

Integral of x^5*cos(x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1             
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 |   5          
 |  x *cos(x) dx
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$$\int\limits_{0}^{1} x^{5} \cos{\left(x \right)}\, dx$$
Integral(x^5*cos(x), (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. The integral of cosine is sine:

    Now evaluate the sub-integral.

  2. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. The integral of sine is negative cosine:

    Now evaluate the sub-integral.

  3. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. The integral of cosine is sine:

    Now evaluate the sub-integral.

  4. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. The integral of sine is negative cosine:

    Now evaluate the sub-integral.

  5. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. The integral of cosine is sine:

    Now evaluate the sub-integral.

  6. The integral of a constant times a function is the constant times the integral of the function:

    1. The integral of sine is negative cosine:

    So, the result is:

  7. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                                    
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 |  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)
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$$\int x^{5} \cos{\left(x \right)}\, dx = C + x^{5} \sin{\left(x \right)} + 5 x^{4} \cos{\left(x \right)} - 20 x^{3} \sin{\left(x \right)} - 60 x^{2} \cos{\left(x \right)} + 120 x \sin{\left(x \right)} + 120 \cos{\left(x \right)}$$
The graph
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
Integral of x^5*cos(x) dx

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