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

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

integral of (sin^4(x))/(cos^4(x))

\int \frac{sin ^{4} ( x )}{cos ^{4} ( x )} d x

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

Solution steps

\int \frac{sin ^{4} ( x )}{cos ^{4} ( x )} d x

Rewrite using trig identities

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

Apply Integral Reduction

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

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

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

Simplify

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

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

Add a constant to the solution

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

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