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

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

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

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

\frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - x tan^{4} (x) + \frac{1}{3} tan^{3} (x) + x - tan (x) + C

Solution steps

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

Apply Integration By Parts

: \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - \int 4 x tan^{3} (x) sec^{2} (x) d x

= \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - \int 4 x tan^{3} (x) sec^{2} (x) d x

\int 4 x tan^{3} (x) sec^{2} (x) d x = x tan^{4} (x) - \frac{1}{3} tan^{3} (x) - x + tan (x)

= \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - (x tan^{4} (x) - \frac{1}{3} tan^{3} (x) - x + tan (x))

Simplify

= \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - x tan^{4} (x) + \frac{1}{3} tan^{3} (x) + x - tan (x)

Add a constant to the solution

= \frac{x sin ^{4} ( x )}{cos ^{4} ( x )} - x tan^{4} (x) + \frac{1}{3} tan^{3} (x) + x - tan (x) + C

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