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

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

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

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

- \frac{1}{2} ln ∣ cos (2 tan (x)) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

= \int tan (2 u) d u

Rewrite using trig identities

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

Apply u-substitution

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - \frac{1}{2} ln ∣ v ∣

Substitute back

= - \frac{1}{2} ln ∣ cos (2 tan (x)) ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ cos (2 tan (x)) ∣ + C

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