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受欢迎的微积分 >

integral of (sin(x))/(cos^2(x)+3)

解答

\int \frac{sin ( x )}{cos ^{2} ( x ) + 3} d x

解答

- \frac{1}{3} arctan (\frac{cos ( x )}{3}) + C

求解步骤

\int \frac{sin ( x )}{cos ^{2} ( x ) + 3} d x

使用换元积分法

= \int - \frac{1}{u ^{2} + 3} d u

提出常数:

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

= - \int \frac{1}{u ^{2} + 3} d u

使用换元积分法

= - \int \frac{1}{3 ( v ^{2} + 1 )} d v

提出常数:

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

= - \frac{1}{3} \cdot \int \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

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

= - \frac{1}{3} arctan (v)

代回

= - \frac{1}{3} arctan (\frac{cos ( x )}{3})

解答补常数

= - \frac{1}{3} arctan (\frac{cos ( x )}{3}) + C

作图

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