integral of xsin(x)sin(x)

解答

\int x sin (x) sin (x) d x

\frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) + C

求解步骤

\int x sin (x) sin (x) d x

化简

x sin (x) sin (x) : x sin^{2} (x)

= \int x sin^{2} (x) d x

使用分布积分法

= \frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \int \frac{1}{2} (x - \frac{1}{2} sin (2 x)) d x

\int \frac{1}{2} (x - \frac{1}{2} sin (2 x)) d x = \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))

= \frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))

解答补常数

= \frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) + C

\int \frac{( 4 + u )}{u} d u

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