integral of (x^2+3x)(sin(x^2))

解答

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

\frac{1}{3} x^{3} sin (x^{2}) + \frac{3}{2} x^{2} sin (x^{2}) - \frac{1}{3} x^{3} sin (x^{2}) - \frac{1}{2} x cos (x^{2}) + \frac{π}{2 2} C (\frac{2}{π} x^{2}) - \frac{3}{2} x^{2} sin (x^{2}) - \frac{3}{2} cos (x^{2}) + C

求解步骤

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

使用分布积分法

= sin (x^{2}) (\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2}) - \int \frac{2}{3} x^{4} cos (x^{2}) + 3 x^{3} cos (x^{2}) d x

\int \frac{2}{3} x^{4} cos (x^{2}) + 3 x^{3} cos (x^{2}) d x = \frac{1}{6} (2 x^{3} sin (x^{2}) - 3 (- x cos (x^{2}) + \frac{π}{2} C (\frac{2}{π} x^{2}))) + \frac{3}{2} (x^{2} sin (x^{2}) + cos (x^{2}))

= sin (x^{2}) (\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2}) - (\frac{1}{6} (2 x^{3} sin (x^{2}) - 3 (- x cos (x^{2}) + \frac{π}{2} C (\frac{2}{π} x^{2}))) + \frac{3}{2} (x^{2} sin (x^{2}) + cos (x^{2})))

化简

sin (x^{2}) (\frac{x ^{3}}{3} + \frac{3 x ^{2}}{2}) - (\frac{1}{6} (2 x^{3} sin (x^{2}) - 3 (- x cos (x^{2}) + \frac{π}{2} C (\frac{2}{π} x^{2}))) + \frac{3}{2} (x^{2} sin (x^{2}) + cos (x^{2}))) : \frac{1}{3} x^{3} sin (x^{2}) + \frac{3}{2} x^{2} sin (x^{2}) - \frac{1}{3} x^{3} sin (x^{2}) - \frac{1}{2} x cos (x^{2}) + \frac{π}{2 2} C (\frac{2}{π} x^{2}) - \frac{3}{2} x^{2} sin (x^{2}) - \frac{3}{2} cos (x^{2})

= \frac{1}{3} x^{3} sin (x^{2}) + \frac{3}{2} x^{2} sin (x^{2}) - \frac{1}{3} x^{3} sin (x^{2}) - \frac{1}{2} x cos (x^{2}) + \frac{π}{2 2} C (\frac{2}{π} x^{2}) - \frac{3}{2} x^{2} sin (x^{2}) - \frac{3}{2} cos (x^{2})

解答补常数

= \frac{1}{3} x^{3} sin (x^{2}) + \frac{3}{2} x^{2} sin (x^{2}) - \frac{1}{3} x^{3} sin (x^{2}) - \frac{1}{2} x cos (x^{2}) + \frac{π}{2 2} C (\frac{2}{π} x^{2}) - \frac{3}{2} x^{2} sin (x^{2}) - \frac{3}{2} cos (x^{2}) + C

\int \frac{x + \frac{1}{2}}{( x ^{2} - x + \frac{5}{4} )} d x

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