What is the integral of (sec^2(sqrt(z)))/(sqrt(z)) ?

The integral of (sec^2(sqrt(z)))/(sqrt(z)) is 2tan(sqrt(z))+C

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integral of (sec^2(sqrt(z)))/(sqrt(z))

\int \frac{sec ^{2} ( z )}{z} d z

2 tan (z) + C

Solution steps

\int \frac{sec ^{2} ( z )}{z} d z

Rewrite using trig identities

= \int \frac{1 + tan ^{2} ( z )}{z} d z

Expand

\frac{1 + tan ^{2} ( z )}{z} : \frac{1}{z} + \frac{tan ^{2} ( z )}{z}

= \int \frac{1}{z} + \frac{tan ^{2} ( z )}{z} d z

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{z} d z + \int \frac{tan ^{2} ( z )}{z} d z

\int \frac{1}{z} d z = 2 z

\int \frac{tan ^{2} ( z )}{z} d z = - 2 z + 2 tan (z)

= 2 z - 2 z + 2 tan (z)

Simplify

= 2 tan (z)

Add a constant to the solution

= 2 tan (z) + C

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What is the integral of (sec^2(sqrt(z)))/(sqrt(z)) ?
The integral of (sec^2(sqrt(z)))/(sqrt(z)) is 2tan(sqrt(z))+C