What is the integral of sec(sqrt(x)) 1/(sqrt(x)) ?

The integral of sec(sqrt(x)) 1/(sqrt(x)) is 2ln|tan(sqrt(x))+sec(sqrt(x))|+C

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integral of sec(sqrt(x)) 1/(sqrt(x))

\int sec (x) \frac{1}{x} d x

2 ln tan (x) + sec (x) + C

Solution steps

\int sec (x) \frac{1}{x} d x

Rewrite using trig identities

= \int \frac{1}{x cos ( x )} d x

Apply u-substitution

= \int 2 sec (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) d u

Use the common integral:

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

= 2 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = x

= 2 ln tan (x) + sec (x)

Add a constant to the solution

= 2 ln tan (x) + sec (x) + C

What is the integral of sec(sqrt(x)) 1/(sqrt(x)) ?
The integral of sec(sqrt(x)) 1/(sqrt(x)) is 2ln|tan(sqrt(x))+sec(sqrt(x))|+C