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

The integral of tan^4(2x)sec^2(2x) is 1/10 tan^5(2x)+C

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

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

\frac{1}{10} tan^{5} (2 x) + C

Solution steps

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

Apply u-substitution

: \int tan^{4} (u) sec^{2} (u) \frac{1}{2} d u

= \int tan^{4} (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^{4} (u) sec^{2} (u) d u

Rewrite using trig identities

= \frac{1}{2} \cdot \int (1 + tan^{2} (u)) tan^{4} (u) d u

Apply u-substitution

: \int v^{4} d v

= \frac{1}{2} \cdot \int v^{4} d v

Apply the Power Rule

: \frac{v ^{5}}{5}

= \frac{1}{2} \cdot \frac{v ^{5}}{5}

Substitute back

= \frac{1}{2} \cdot \frac{tan ^{5} ( 2 x )}{5}

Simplify

\frac{1}{2} \cdot \frac{tan ^{5} ( 2 x )}{5} : \frac{1}{10} tan^{5} (2 x)

= \frac{1}{10} tan^{5} (2 x)

Add a constant to the solution

= \frac{1}{10} tan^{5} (2 x) + C

\int \frac{5 ( tan ( x ) ) ^{2}}{sec ( x )} d x

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\int 90 x + \frac{500}{1 + x} d x

What is the integral of tan^4(2x)sec^2(2x) ?
The integral of tan^4(2x)sec^2(2x) is 1/10 tan^5(2x)+C