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

The integral of (tan(sqrt(x)))/(sqrt(x)) is -2ln|cos(sqrt(x))|+C

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

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

- 2 ln cos (x) + C

Solution steps

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

Apply u-substitution

: \int 2 tan (u) d u

= \int 2 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 tan (u) d u

Rewrite using trig identities

= 2 \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

: \int - \frac{1}{v} d v

= 2 \cdot \int - \frac{1}{v} d v

Take the constant out:

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

= 2 (- \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 2 (- ln ∣ v ∣)

Substitute back

= 2 (- ln cos (x))

Simplify

= - 2 ln cos (x)

Add a constant to the solution

= - 2 ln cos (x) + C

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