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受欢迎的微积分 >

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

解答

\int \frac{2}{1 + cos ^{2} ( x )} d x

解答

2 arctan (\frac{1}{2} tan (x)) + C

求解步骤

\int \frac{2}{1 + cos ^{2} ( x )} d x

提出常数:

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

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

使用三角恒等式改写

= 2 \cdot \int \frac{2}{cos ( 2 x ) + 3} d x

提出常数:

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

= 2 \cdot 2 \cdot \int \frac{1}{cos ( 2 x ) + 3} d x

使用换元积分法

= 2 \cdot 2 \cdot \int \frac{1}{2 ( cos ( u ) + 3 )} d u

提出常数:

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

= 2 \cdot 2 \cdot \frac{1}{2} \cdot \int \frac{1}{cos ( u ) + 3} d u

使用换元积分法

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

使用换元积分法

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

提出常数:

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

= 2 \cdot 2 \cdot \frac{1}{2} \cdot \frac{1}{2} \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)

= 2 \cdot 2 \cdot \frac{1}{2} \cdot \frac{1}{2} arctan (w)

代回

= 2 \cdot 2 \cdot \frac{1}{2} \cdot \frac{1}{2} arctan (\frac{tan ( \frac{2 x}{2} )}{2})

化简

2 \cdot 2 \cdot \frac{1}{2} \cdot \frac{1}{2} arctan (\frac{tan ( \frac{2 x}{2} )}{2}) : 2 arctan (\frac{1}{2} tan (x))

= 2 arctan (\frac{1}{2} tan (x))

解答补常数

= 2 arctan (\frac{1}{2} tan (x)) + C

作图

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