What is the integral from 0 to pi of x^5cos(x) ?

The integral from 0 to pi of x^5cos(x) is -5pi^4+60pi^2-240

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integral from 0 to pi of x^5cos(x)

\int_{0}^{π} x^{5} cos (x) d x

- 5 π^{4} + 60 π^{2} - 240

Decimal

- 134.86919 \dots

Solution steps

\int_{0}^{π} x^{5} cos (x) d x

Apply Integration By Parts

= [x^{5} sin (x) - \int 5 x^{4} sin (x) d x]_{0}^{π}

\int 5 x^{4} sin (x) d x = 5 (- x^{4} cos (x) + 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))))

= [x^{5} sin (x) - 5 (- x^{4} cos (x) + 4 (x^{3} sin (x) - 3 (- x^{2} cos (x) + 2 (x sin (x) + cos (x)))))]_{0}^{π}

Compute the boundaries:

- 5 π^{4} + 60 π^{2} - 240

= - 5 π^{4} + 60 π^{2} - 240

What is the integral from 0 to pi of x^5cos(x) ?
The integral from 0 to pi of x^5cos(x) is -5pi^4+60pi^2-240