What is the integral of sec^3(β) ?

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

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integral of sec^3(β)

\int sec^{3} (β) d β

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

Solution steps

\int sec^{3} (β) d β

Apply Integral Reduction

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

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

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

Simplify

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

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

Add a constant to the solution

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

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