What is the integral from 0 to pi of x^6sin(x) ?

The integral from 0 to pi of x^6sin(x) is pi^6-30pi^4+360pi^2-1440

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integral from 0 to pi of x^6sin(x)

\int_{0}^{π} x^{6} sin (x) d x

π^{6} - 30 π^{4} + 360 π^{2} - 1440

Decimal

152.17404 \dots

Solution steps

\int_{0}^{π} x^{6} sin (x) d x

Apply Integration By Parts

= [- x^{6} cos (x) - \int - 6 x^{5} cos (x) d x]_{0}^{π}

\int - 6 x^{5} cos (x) d x = - 6 (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))))))

= [- x^{6} cos (x) - (- 6 (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}

Simplify

= [- x^{6} cos (x) + 6 (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:

π^{6} - 30 π^{4} + 360 π^{2} - 1440

= π^{6} - 30 π^{4} + 360 π^{2} - 1440

What is the integral from 0 to pi of x^6sin(x) ?
The integral from 0 to pi of x^6sin(x) is pi^6-30pi^4+360pi^2-1440