integral of xsin^2(x)cos^3(x)

解答

\int x sin^{2} (x) cos^{3} (x) d x

\frac{1}{3} x sin^{3} (x) - \frac{1}{5} x sin^{5} (x) + \frac{2}{15} cos (x) + \frac{1}{45} cos^{3} (x) - \frac{1}{25} cos^{5} (x) + C

求解步骤

\int x sin^{2} (x) cos^{3} (x) d x

使用分布积分法

= x (\frac{1}{3} sin^{3} (x) - \frac{1}{5} sin^{5} (x)) - \int \frac{1}{3} sin^{3} (x) - \frac{1}{5} sin^{5} (x) d x

\int \frac{1}{3} sin^{3} (x) - \frac{1}{5} sin^{5} (x) d x = - \frac{2}{15} cos (x) + \frac{1}{9} cos^{3} (x) - \frac{2}{15} cos^{3} (x) + \frac{1}{25} cos^{5} (x)

= x (\frac{1}{3} sin^{3} (x) - \frac{1}{5} sin^{5} (x)) - (- \frac{2}{15} cos (x) + \frac{1}{9} cos^{3} (x) - \frac{2}{15} cos^{3} (x) + \frac{1}{25} cos^{5} (x))

化简

x (\frac{1}{3} sin^{3} (x) - \frac{1}{5} sin^{5} (x)) - (- \frac{2}{15} cos (x) + \frac{1}{9} cos^{3} (x) - \frac{2}{15} cos^{3} (x) + \frac{1}{25} cos^{5} (x)) : \frac{1}{3} x sin^{3} (x) - \frac{1}{5} x sin^{5} (x) + \frac{2}{15} cos (x) + \frac{1}{45} cos^{3} (x) - \frac{1}{25} cos^{5} (x)

= \frac{1}{3} x sin^{3} (x) - \frac{1}{5} x sin^{5} (x) + \frac{2}{15} cos (x) + \frac{1}{45} cos^{3} (x) - \frac{1}{25} cos^{5} (x)

解答补常数

= \frac{1}{3} x sin^{3} (x) - \frac{1}{5} x sin^{5} (x) + \frac{2}{15} cos (x) + \frac{1}{45} cos^{3} (x) - \frac{1}{25} cos^{5} (x) + C

x \to 125 lim (\frac{x - 125}{3 x - 5})

\int (4 x^{2} + 8 x + 5)^{6} (x + 1) d x

\int \frac{1}{( x + 7 ) ( x ^{2} + 16 )} d x

\int \frac{2 x ^{3} + x + 1}{x ^{2} + 9} d x

d er i v a t i v e 1 - 9 x