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受欢迎的微积分 >

integral of 2sec^2(4/5 x)

解答

\int 2 sec^{2} (\frac{4}{5} x) d x

解答

\frac{5}{2} tan (\frac{4 x}{5}) + C

求解步骤

\int 2 sec^{2} (\frac{4}{5} x) d x

提出常数:

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

= 2 \cdot \int sec^{2} (\frac{4}{5} x) d x

使用三角恒等式改写

= 2 \cdot \int \frac{1}{cos ^{2} ( \frac{4 x}{5} )} d x

使用基本三角恒等式:

\frac{1}{cos ( x )} = sec (x)

= 2 \cdot \int sec^{2} (\frac{4 x}{5}) d x

使用三角恒等式改写

= 2 \cdot \int 1 + tan^{2} (\frac{4 x}{5}) d x

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 1 d x + \int tan^{2} (\frac{4 x}{5}) d x)

\int 1 d x = x

\int tan^{2} (\frac{4 x}{5}) d x = - x + \frac{5}{4} tan (\frac{4 x}{5})

= 2 (x - x + \frac{5}{4} tan (\frac{4 x}{5}))

化简

2 (x - x + \frac{5}{4} tan (\frac{4 x}{5})) : \frac{5}{2} tan (\frac{4 x}{5})

= \frac{5}{2} tan (\frac{4 x}{5})

解答补常数

= \frac{5}{2} tan (\frac{4 x}{5}) + C

作图

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\int cos (x) \frac{1}{x} d x

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\int tan (x) sec^{\frac{5}{4}} (x) d x

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