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

Lösung

\int_{0}^{π} sin (x) x^{5} d x

π^{5} - 20 (π^{3} - 6 π)

Dezimale

62.88526 \dots

Schritte zur Lösung

\int_{0}^{π} sin (x) x^{5} d x

Wende die partielle Integration an

= [- 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}

Vereinfache

= [- 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}

Berechne die Grenzen:

π^{5} - 20 (π^{3} - 6 π)

= π^{5} - 20 (π^{3} - 6 π)

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

d er i v a t i v e 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

d er i v a t i v e f (x) = x + x + 5