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

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

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integral of (tan^2(x))/(sec^2(x))

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

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

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int 1 - cos (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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

\int \frac{2 x + 1}{x - 2} d x

y^{^{'}} = y (y^{3} - 1)

\int (e^{- x} - e^{5}) \cdot (e^{x} + e^{- 2}) d x

y^{^{'}} - y = 10 t e^{2 t}, y (0) = 1

f (x) = (- 25 x^{2} + 36)^{2}

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