integral of 19x^2sin(pix)

解答

\int 19 x^{2} sin (π x) d x

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

求解步骤

\int 19 x^{2} sin (π x) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 19 \cdot \int x^{2} sin (π x) d x

使用分布积分法

= 19 (- \frac{1}{π} x^{2} cos (π x) - \int - \frac{2}{π} x cos (π x) d x)

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

= 19 (- \frac{1}{π} x^{2} cos (π x) - (- \frac{2}{π} (\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x))))

化简

= 19 (- \frac{1}{π} x^{2} cos (π x) + \frac{2}{π} (\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x)))

解答补常数

= 19 (- \frac{1}{π} x^{2} cos (π x) + \frac{2}{π} (\frac{1}{π} x sin (π x) + \frac{1}{π ^{2}} cos (π x))) + C

x \to - 4 lim (2 x + 1)

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