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

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

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

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

- tan (x) + 2 x + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{- 1 + 2 cos ^{2} ( x )}{cos ^{2} ( x )} : - \frac{1}{cos ^{2} ( x )} + 2

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

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

\int 2 d x = 2 x

= - tan (x) + 2 x

Add a constant to the solution

= - tan (x) + 2 x + C

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\int_{0}^{1} [(x^{2} + 2) - (- x)] d x

\int x (x - 3)^{7} d x

\int_{1}^{t} \frac{33}{x ^{3}} d x

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