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

The integral of x/(cos^2(2x)) is 1/2 xtan(2x)+1/4 ln|cos(2x)|+C

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integral of x/(cos^2(2x))

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

\frac{1}{2} x tan (2 x) + \frac{1}{4} ln ∣ cos (2 x) ∣ + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{1}{2} tan (2 x) d x = - \frac{1}{4} ln ∣ cos (2 x) ∣

= \frac{1}{2} x tan (2 x) - (- \frac{1}{4} ln ∣ cos (2 x) ∣)

Simplify

= \frac{1}{2} x tan (2 x) + \frac{1}{4} ln ∣ cos (2 x) ∣

Add a constant to the solution

= \frac{1}{2} x tan (2 x) + \frac{1}{4} ln ∣ cos (2 x) ∣ + C

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What is the integral of x/(cos^2(2x)) ?
The integral of x/(cos^2(2x)) is 1/2 xtan(2x)+1/4 ln|cos(2x)|+C