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

e^{x} \frac{d y}{d x} + 2 e^{x} y = 1

\frac{d}{d x} (c_{1} e^{- 2 x} + c_{2} x e^{- 2 x})

f (x) = e^{\frac{1}{x}} - \frac{e ^{\frac{1}{x}}}{x}

\int \frac{1}{( 5 - 4 x - x ^{2} ) ^{\frac{5}{2}}} d x

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

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