integral of (3x^2+5)cos(7x)

解答

\int (3 x^{2} + 5) cos (7 x) d x

\frac{3}{7} x^{2} sin (7 x) + \frac{6}{49} x cos (7 x) + \frac{239}{343} sin (7 x) + C

求解步骤

\int (3 x^{2} + 5) cos (7 x) d x

乘开

(3 x^{2} + 5) cos (7 x) : 3 x^{2} cos (7 x) + 5 cos (7 x)

= \int 3 x^{2} cos (7 x) + 5 cos (7 x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 3 x^{2} cos (7 x) d x + \int 5 cos (7 x) d x

\int 3 x^{2} cos (7 x) d x = \frac{3}{343} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x)))

\int 5 cos (7 x) d x = \frac{5}{7} sin (7 x)

= \frac{3}{343} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x))) + \frac{5}{7} sin (7 x)

化简

\frac{3}{343} (49 x^{2} sin (7 x) - 2 (- 7 x cos (7 x) + sin (7 x))) + \frac{5}{7} sin (7 x) : \frac{3}{7} x^{2} sin (7 x) + \frac{6}{49} x cos (7 x) + \frac{239}{343} sin (7 x)

= \frac{3}{7} x^{2} sin (7 x) + \frac{6}{49} x cos (7 x) + \frac{239}{343} sin (7 x)

解答补常数

= \frac{3}{7} x^{2} sin (7 x) + \frac{6}{49} x cos (7 x) + \frac{239}{343} sin (7 x) + C

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