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

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

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integral of (1+sin(2x))/(cos^2(2x))

\int \frac{1 + sin ( 2 x )}{cos ^{2} ( 2 x )} d x

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

Solution steps

\int \frac{1 + sin ( 2 x )}{cos ^{2} ( 2 x )} d x

Rewrite using trig identities

= \int (1 + sin (2 x)) sec^{2} (2 x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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