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受欢迎的微积分 >

integral of x/(1+x^2y^2)

解答

\int \frac{x}{1 + x ^{2} y ^{2}} d y

解答

arctan (x y) + C

求解步骤

\int \frac{x}{1 + x ^{2} y ^{2}} d y

提出常数:

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

= x \cdot \int \frac{1}{1 + x ^{2} y ^{2}} d y

使用换元积分法

= x \cdot \int \frac{1}{x ( 1 + u ^{2} )} d u

提出常数:

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

= x \frac{1}{x} \cdot \int \frac{1}{1 + u ^{2}} d u

使用积分常见定则:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= x \frac{1}{x} arctan (u)

u = x y

代回

= x \frac{1}{x} arctan (x y)

化简

x \frac{1}{x} arctan (x y) : arctan (x y)

= arctan (x y)

解答补常数

= arctan (x y) + C

作图

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