What is the integral of sec^3(y) ?

The integral of sec^3(y) is 1/2 sec(y)tan(y)+1/2 ln|tan(y)+sec(y)|+C

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integral of sec^3(y)

\int sec^{3} (y) d y

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

Solution steps

\int sec^{3} (y) d y

Apply Integral Reduction

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

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

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of sec^3(y) ?
The integral of sec^3(y) is 1/2 sec(y)tan(y)+1/2 ln|tan(y)+sec(y)|+C