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

The integral from 0 to pi of sin(x)x^5 is pi^5-20(pi^3-6pi)

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

\int_{0}^{π} sin (x) x^{5} d x

π^{5} - 20 (π^{3} - 6 π)

Decimal Notation

62.88526 \dots

Solution steps

\int_{0}^{π} sin (x) x^{5} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Compute the boundaries:

π^{5} - 20 (π^{3} - 6 π)

= π^{5} - 20 (π^{3} - 6 π)

x \to - 2 - lim (\frac{- 4 x}{x + 2})

\int x (3 x - 1)^{10} d x

f (x) = \frac{6 x}{x ^{2} + 5}

\int sin^{3} (11 x) cos^{3} (11 x) d x

\int (9 - x^{2}) d x

What is the integral from 0 to pi of sin(x)x^5 ?
The integral from 0 to pi of sin(x)x^5 is pi^5-20(pi^3-6pi)