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

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

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

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

tan (x) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Rewrite using trig identities

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

Expand

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

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

Apply the Sum Rule:

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

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

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

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

= \frac{1}{2} (2 x - 2 x + 2 tan (x))

Simplify

\frac{1}{2} (2 x - 2 x + 2 tan (x)) : tan (x)

= tan (x)

Add a constant to the solution

= tan (x) + C

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