What is the integral of (tan^2(2x)+tan^4(2x)) ?

The integral of (tan^2(2x)+tan^4(2x)) is 1/6 tan^3(2x)+C

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integral of (tan^2(2x)+tan^4(2x))

\int (tan^{2} (2 x) + tan^{4} (2 x)) d x

\frac{1}{6} tan^{3} (2 x) + C

Solution steps

\int tan^{2} (2 x) + tan^{4} (2 x) d x

Simplify

tan^{2} (2 x) + tan^{4} (2 x) : tan^{2} (2 x) sec^{2} (2 x)

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

Apply u-substitution

: \int tan^{2} (u) sec^{2} (u) \frac{1}{2} d u

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

Take the constant out:

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

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

Rewrite using trig identities

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

Apply u-substitution

: \int v^{2} d v

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

Apply the Power Rule

: \frac{v ^{3}}{3}

= \frac{1}{2} \cdot \frac{v ^{3}}{3}

Substitute back

= \frac{1}{2} \cdot \frac{tan ^{3} ( 2 x )}{3}

Simplify

\frac{1}{2} \cdot \frac{tan ^{3} ( 2 x )}{3} : \frac{1}{6} tan^{3} (2 x)

= \frac{1}{6} tan^{3} (2 x)

Add a constant to the solution

= \frac{1}{6} tan^{3} (2 x) + C

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What is the integral of (tan^2(2x)+tan^4(2x)) ?
The integral of (tan^2(2x)+tan^4(2x)) is 1/6 tan^3(2x)+C