What is the integral of (sin(2t))/(cos^2(2t)) ?

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

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integral of (sin(2t))/(cos^2(2t))

\int \frac{sin ( 2 t )}{cos ^{2} ( 2 t )} d t

\frac{1}{2} cos (2 t) + \frac{1}{2} sin (2 t) tan (2 t) + C

Solution steps

\int \frac{sin ( 2 t )}{cos ^{2} ( 2 t )} d t

Apply Integration By Parts

: - \int sin (2 t) d t + \frac{1}{2} sin (2 t) tan (2 t)

= - \int sin (2 t) d t + \frac{1}{2} sin (2 t) tan (2 t)

\int sin (2 t) d t = - \frac{1}{2} cos (2 t)

= - (- \frac{1}{2} cos (2 t)) + \frac{1}{2} sin (2 t) tan (2 t)

Simplify

= \frac{1}{2} cos (2 t) + \frac{1}{2} sin (2 t) tan (2 t)

Add a constant to the solution

= \frac{1}{2} cos (2 t) + \frac{1}{2} sin (2 t) tan (2 t) + C

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