What is the integral of sec^5(u) ?

The integral of sec^5(u) is (sec^4(u)sin(u))/4+3/4 (1/2 sec(u)tan(u)+1/2 ln|tan(u)+sec(u)|)+C

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integral of sec^5(u)

\int sec^{5} (u) d u

\frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣) + C

Solution steps

\int sec^{5} (u) d u

Apply Integral Reduction

= \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} \cdot \int sec^{3} (u) d u

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

= \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Add a constant to the solution

= \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣) + C

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What is the integral of sec^5(u) ?
The integral of sec^5(u) is (sec^4(u)sin(u))/4+3/4 (1/2 sec(u)tan(u)+1/2 ln|tan(u)+sec(u)|)+C