integral of x^9sin(x)

解答

\int x^{9} sin (x) d x

- x^{9} cos (x) + 9 (x^{8} sin (x) - 8 (- x^{7} cos (x) + 7 (x^{6} sin (x) - 6 (- 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))))))))) + C

求解步骤

\int x^{9} sin (x) d x

使用分布积分法

= - x^{9} cos (x) - \int - 9 x^{8} cos (x) d x

\int - 9 x^{8} cos (x) d x = - 9 (x^{8} sin (x) - 8 (- x^{7} cos (x) + 7 (x^{6} sin (x) - 6 (- 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)))))))))

= - x^{9} cos (x) - (- 9 (x^{8} sin (x) - 8 (- x^{7} cos (x) + 7 (x^{6} sin (x) - 6 (- 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))))))))))

化简

= - x^{9} cos (x) + 9 (x^{8} sin (x) - 8 (- x^{7} cos (x) + 7 (x^{6} sin (x) - 6 (- 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)))))))))

解答补常数

= - x^{9} cos (x) + 9 (x^{8} sin (x) - 8 (- x^{7} cos (x) + 7 (x^{6} sin (x) - 6 (- 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))))))))) + C

\int e^{θ} csc (e^{θ} + 1) d θ

\int e^{\frac{x}{4}} sin (\frac{x}{8}) cos (\frac{x}{8}) d x

\int (1 + 2 x)^{5} d x

\int \frac{x}{( 5 x ^{2} - 4 ) ^{4}} d x

\int \frac{sin ^{5} ( x ) - cos ^{2} ( x )}{cos ( x )} d x