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

解答

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

- x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

求解步骤

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

使用三角恒等式改写

= \int \frac{1 - cos ^{2} ( x )}{cos ^{3} ( x )} d x

乘开

\frac{1 - cos ^{2} ( x )}{cos ^{3} ( x )} : \frac{1}{cos ^{3} ( x )} - \frac{1}{cos ( x )}

= \int \frac{1}{cos ^{3} ( x )} - \frac{1}{cos ( 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}{cos ^{3} ( x )} d x - \int \frac{1}{cos ( x )} d x

\int \frac{1}{cos ^{3} ( x )} d x = - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

= - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ - ln ∣ tan (x) + sec (x) ∣

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