integral of ((sec^2(sqrt(x))))/(sqrt(x))

解答

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

2 tan (x) + C

求解步骤

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

使用三角恒等式改写

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

乘开

\frac{1 + tan ^{2} ( x )}{x} : \frac{1}{x} + \frac{tan ^{2} ( x )}{x}

= \int \frac{1}{x} + \frac{tan ^{2} ( x )}{x} 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

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

\int \frac{1}{x} d x = 2 x

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

= 2 x - 2 x + 2 tan (x)

化简

= 2 tan (x)

解答补常数

= 2 tan (x) + C

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