integral from 0 to x^3 of sin(y/x)

解答

\int_{0}^{x^{3}} sin (\frac{y}{x}) d y

x (- cos (x^{2}) + 1)

求解步骤

\int_{0}^{x^{3}} sin (\frac{y}{x}) d y

使用换元积分法

= \int_{0}^{x^{2}} sin (u) x d u

提出常数:

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

= x \cdot \int_{0}^{x^{2}} sin (u) d u

使用积分常见定则:

\int sin (u) d u = - cos (u)

= x [- cos (u)]_{0}^{x^{2}}

带入上下限计算后得:

- cos (x^{2}) + 1

= x (- cos (x^{2}) + 1)

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