What is the integral of cos(2x)tan(x) ?

The integral of cos(2x)tan(x) is 1/2 tan(x)sin(2x)+1/2 ln(cos^2(x))+C

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integral of cos(2x)tan(x)

\int cos (2 x) tan (x) d x

\frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x)) + C

Solution steps

\int cos (2 x) tan (x) d x

Apply Integration By Parts

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

\int sec^{2} (x) \frac{1}{2} sin (2 x) d x = - \frac{1}{2} ln (cos^{2} (x))

= tan (x) \frac{1}{2} sin (2 x) - (- \frac{1}{2} ln (cos^{2} (x)))

Simplify

= \frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x))

Add a constant to the solution

= \frac{1}{2} tan (x) sin (2 x) + \frac{1}{2} ln (cos^{2} (x)) + C

What is the integral of cos(2x)tan(x) ?
The integral of cos(2x)tan(x) is 1/2 tan(x)sin(2x)+1/2 ln(cos^2(x))+C