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

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

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integral of 1-1/(cos^2(2x))

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

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

Solution steps

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

Use the basic trigonometric identity:

\frac{1}{cos ( x )} = sec (x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d x = x

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

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

Add a constant to the solution

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

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