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

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

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integral of (sin^2(x)-cos^2(x))

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

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

Solution steps

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

Rewrite using trig identities

= \int 1 - 2 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

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

\int 1 d x = x

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

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

Simplify

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

Add a constant to the solution

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

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