What is the integral of 1/((1-sin^2(x))^{3/2)} ?

The integral of 1/((1-sin^2(x))^{3/2)} is 1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|+C

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integral of 1/((1-sin^2(x))^{3/2)}

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

\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

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

Simplify

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

= \int sec^{3} (x) d x

Apply Integral Reduction

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

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

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

Simplify

\frac{sec ^{2} ( x ) sin ( x )}{2} + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ : \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

= \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

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What is the integral of 1/((1-sin^2(x))^{3/2)} ?
The integral of 1/((1-sin^2(x))^{3/2)} is 1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(x)|+C