integral from 0 to 1 of x*sin(pix)

解答

\int_{0}^{1} x \cdot sin (π x) d x

\frac{1}{π}

十进制

0.31830 \dots

求解步骤

\int_{0}^{1} x sin (π x) d x

使用分布积分法

= [- \frac{1}{π} x cos (π x) - \int - \frac{1}{π} cos (π x) d x]_{0}^{1}

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

= [- \frac{1}{π} x cos (π x) - (- \frac{1}{π ^{2}} sin (π x))]_{0}^{1}

化简

= [- \frac{1}{π} x cos (π x) + \frac{1}{π ^{2}} sin (π x)]_{0}^{1}

带入上下限计算后得:

\frac{1}{π}

= \frac{1}{π}

d er i v a t i v e y = sec (3 x^{4} + 5)

θ \to 0 lim (\frac{cos ( θ ) - 1}{θ})

\int_{0}^{2} π (\frac{1}{x + 4})^{2} d x

d er i v a t i v e f (x) = ln (81 sin^{2} (x))

\frac{d}{d x} (\frac{2 x + 1}{3 x - 2})