What is the integral of sec^2(y)tan(tan(y)) ?

The integral of sec^2(y)tan(tan(y)) is -ln|cos(tan(y))|+C

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integral of sec^2(y)tan(tan(y))

integral

sec^{2} (y) tan (tan (y))

- ln ∣ cos (tan (y)) ∣ + C

Solution steps

\int sec^{2} (y) tan (tan (y)) d y

Rewrite using trig identities

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

Apply u-substitution

= \int tan (u) d u

Rewrite using trig identities

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

Apply u-substitution

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - ln ∣ v ∣

Substitute back

= - ln ∣ cos (tan (y)) ∣

Add a constant to the solution

= - ln ∣ cos (tan (y)) ∣ + C

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