integral of x^8sin(x)

解答

\int x^{8} sin (x) d x

- x^{8} cos (x) + 8 (x^{7} sin (x) - 7 (- 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)))))))) + C

求解步骤

\int x^{8} sin (x) d x

使用分布积分法

= - x^{8} cos (x) - \int - 8 x^{7} cos (x) d x

\int - 8 x^{7} cos (x) d x = - 8 (x^{7} sin (x) - 7 (- 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))))))))

= - x^{8} cos (x) - (- 8 (x^{7} sin (x) - 7 (- 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)))))))))

化简

= - x^{8} cos (x) + 8 (x^{7} sin (x) - 7 (- 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))))))))

解答补常数

= - x^{8} cos (x) + 8 (x^{7} sin (x) - 7 (- 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)))))))) + C

\int sin (\frac{nπ x}{10}) d x

\int \frac{y}{5 - y} d y

\int \frac{3 x ^{2} + x - 2}{x ^{4} + 8 x ^{2} + 16} d x

\int 8 x y^{2} d x

\int \frac{1}{x ( x - 1 ) ( x + 1 )} d x