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受欢迎的微积分 >

integral of y^2e^{xy^2}

解答

\int y^{2} e^{x y^{2}} d x

解答

e^{y^{2} x} + C

求解步骤

\int y^{2} e^{x y^{2}} d x

提出常数:

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

= y^{2} \cdot \int e^{y^{2} x} d x

使用换元积分法

= y^{2} \cdot \int \frac{e ^{u}}{y ^{2}} d u

提出常数:

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

= y^{2} \frac{1}{y ^{2}} \cdot \int e^{u} d u

使用积分常见定则:

\int e^{u} d u = e^{u}

= y^{2} \frac{1}{y ^{2}} e^{u}

u = y^{2} x

代回

= y^{2} \frac{1}{y ^{2}} e^{y^{2} x}

化简

y^{2} \frac{1}{y ^{2}} e^{y^{2} x} : e^{y^{2} x}

= e^{y^{2} x}

解答补常数

= e^{y^{2} x} + C

作图

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