What is the integral of tan^4(5x) ?

The integral of tan^4(5x) is 1/15 tan^3(5x)+x-1/5 tan(5x)+C

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integral of tan^4(5x)

\int tan^{4} (5 x) d x

\frac{1}{15} tan^{3} (5 x) + x - \frac{1}{5} tan (5 x) + C

Solution steps

\int tan^{4} (5 x) d x

Apply u-substitution

= \int tan^{4} (u) \frac{1}{5} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int tan^{4} (u) d u

Apply Integral Reduction

= \frac{1}{5} (\frac{tan ^{3} ( u )}{3} - \int tan^{2} (u) d u)

\int tan^{2} (u) d u = - u + tan (u)

= \frac{1}{5} (\frac{tan ^{3} ( u )}{3} - (- u + tan (u)))

Substitute back

u = 5 x

= \frac{1}{5} (\frac{tan ^{3} ( 5 x )}{3} - (- 5 x + tan (5 x)))

Simplify

\frac{1}{5} (\frac{tan ^{3} ( 5 x )}{3} - (- 5 x + tan (5 x))) : \frac{1}{15} tan^{3} (5 x) + x - \frac{1}{5} tan (5 x)

= \frac{1}{15} tan^{3} (5 x) + x - \frac{1}{5} tan (5 x)

Add a constant to the solution

= \frac{1}{15} tan^{3} (5 x) + x - \frac{1}{5} tan (5 x) + C

What is the integral of tan^4(5x) ?
The integral of tan^4(5x) is 1/15 tan^3(5x)+x-1/5 tan(5x)+C