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

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

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integral of tan^4(x/4)

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

\frac{4}{3} tan^{3} (\frac{x}{4}) + x - 4 tan (\frac{x}{4}) + C

Solution steps

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

Apply u-substitution

: \int tan^{4} (u) \cdot 4 d u

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

Take the constant out:

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

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

Apply Integral Reduction

= 4 (\frac{tan ^{3} ( u )}{3} - \int tan^{2} (u) d u)

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

= 4 (\frac{tan ^{3} ( u )}{3} - (- u + tan (u)))

Substitute back

u = \frac{x}{4}

= 4 (\frac{tan ^{3} ( \frac{x}{4} )}{3} - (- \frac{x}{4} + tan (\frac{x}{4})))

Simplify

4 (\frac{tan ^{3} ( \frac{x}{4} )}{3} - (- \frac{x}{4} + tan (\frac{x}{4}))) : \frac{4}{3} tan^{3} (\frac{x}{4}) + x - 4 tan (\frac{x}{4})

= \frac{4}{3} tan^{3} (\frac{x}{4}) + x - 4 tan (\frac{x}{4})

Add a constant to the solution

= \frac{4}{3} tan^{3} (\frac{x}{4}) + x - 4 tan (\frac{x}{4}) + C

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