What is the integral of (tan(sqrt(y)))/(2sqrt(y)) ?

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

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integral of (tan(sqrt(y)))/(2sqrt(y))

\int \frac{tan ( y )}{2 y} d y

- ln ∣ cos (y) ∣ + C

Solution steps

\int \frac{tan ( y )}{2 y} d y

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{tan ( y )}{y} d y

Apply u-substitution

= \frac{1}{2} \cdot \int 2 tan (u) d u

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot \int tan (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot 2 \cdot \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

= \frac{1}{2} \cdot 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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{2} \cdot 2 (- ln ∣ v ∣)

Substitute back

= \frac{1}{2} \cdot 2 (- ln ∣ cos (y) ∣)

Simplify

\frac{1}{2} \cdot 2 (- ln ∣ cos (y) ∣) : - ln ∣ cos (y) ∣

= - ln ∣ cos (y) ∣

Add a constant to the solution

= - ln ∣ cos (y) ∣ + C

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