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受欢迎的微积分 >

integral of (sin(2x))/(cos^2(2x)+5)

解答

\int \frac{sin ( 2 x )}{cos ^{2} ( 2 x ) + 5} d x

解答

- \frac{1}{2 5} arctan (\frac{1}{5} cos (2 x)) + C

求解步骤

\int \frac{sin ( 2 x )}{cos ^{2} ( 2 x ) + 5} d x

使用换元积分法

= \int \frac{sin ( u )}{2 ( cos ^{2} ( u ) + 5 )} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{sin ( u )}{cos ^{2} ( u ) + 5} d u

使用换元积分法

= \frac{1}{2} \cdot \int - \frac{1}{v ^{2} + 5} d v

提出常数:

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

= \frac{1}{2} (- \int \frac{1}{v ^{2} + 5} d v)

使用换元积分法

= \frac{1}{2} (- \int \frac{1}{5 ( w ^{2} + 1 )} d w)

提出常数:

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

= \frac{1}{2} (- \frac{1}{5} \cdot \int \frac{1}{w ^{2} + 1} d w)

使用积分常见定则:

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

= \frac{1}{2} (- \frac{1}{5} arctan (w))

代回

= \frac{1}{2} (- \frac{1}{5} arctan (\frac{cos ( 2 x )}{5}))

化简

\frac{1}{2} (- \frac{1}{5} arctan (\frac{cos ( 2 x )}{5})) : - \frac{1}{2 5} arctan (\frac{1}{5} cos (2 x))

= - \frac{1}{2 5} arctan (\frac{1}{5} cos (2 x))

解答补常数

= - \frac{1}{2 5} arctan (\frac{1}{5} cos (2 x)) + C

作图

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