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受欢迎的微积分 >

integral of 2tan^2(x)sec^4(x)

解答

\int 2 tan^{2} (x) sec^{4} (x) d x

解答

2 (\frac{tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5}) + C

求解步骤

\int 2 tan^{2} (x) sec^{4} (x) d x

提出常数:

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

= 2 \cdot \int tan^{2} (x) sec^{4} (x) d x

使用三角恒等式改写

= 2 \cdot \int (1 + tan^{2} (x)) sec^{2} (x) tan^{2} (x) d x

使用换元积分法

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

乘开

u^{2} (1 + u^{2}) : u^{2} + u^{4}

= 2 \cdot \int u^{2} + u^{4} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int u^{2} d u + \int u^{4} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int u^{4} d u = \frac{u ^{5}}{5}

= 2 (\frac{u ^{3}}{3} + \frac{u ^{5}}{5})

u = tan (x)

代回

= 2 (\frac{tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5})

解答补常数

= 2 (\frac{tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5}) + C

作图

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