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

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

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integral of (sec(sqrt(x))tan(sqrt(x)))/(sqrt(x))

\int \frac{sec ( x ) tan ( x )}{x} d x

2 sec (x) + C

Solution steps

\int \frac{sec ( x ) tan ( x )}{x} d x

Apply u-substitution

: \int 2 sec (u) tan (u) d u

= \int 2 sec (u) tan (u) d u

Take the constant out:

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

= 2 \cdot \int sec (u) tan (u) d u

Apply u-substitution

: \int 1 d v

= 2 \cdot \int 1 d v

Integral of a constant:

\int a d x = a x

= 2 \cdot 1 \cdot v

Substitute back

= 2 \cdot 1 \cdot sec (x)

Simplify

= 2 sec (x)

Add a constant to the solution

= 2 sec (x) + C

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