What is the integral of-sin^2(x)+tan^2(x) ?

The integral of-sin^2(x)+tan^2(x) is -3/2 x+1/4 sin(2x)+tan(x)+C

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integral of-sin^2(x)+tan^2(x)

integral

- sin^{2} (x) + tan^{2} (x)

- \frac{3}{2} x + \frac{1}{4} sin (2 x) + tan (x) + C

Solution steps

\int - sin^{2} (x) + tan^{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 sin^{2} (x) d x + \int tan^{2} (x) d x

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

\int tan^{2} (x) d x = - x + tan (x)

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

Simplify

- \frac{1}{2} (x - \frac{1}{2} sin (2 x)) - x + tan (x) : - \frac{3}{2} x + \frac{1}{4} sin (2 x) + tan (x)

= - \frac{3}{2} x + \frac{1}{4} sin (2 x) + tan (x)

Add a constant to the solution

= - \frac{3}{2} x + \frac{1}{4} sin (2 x) + tan (x) + C

What is the integral of-sin^2(x)+tan^2(x) ?
The integral of-sin^2(x)+tan^2(x) is -3/2 x+1/4 sin(2x)+tan(x)+C