What is the integral of sin^2(ax) ?

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

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integral of sin^2(ax)

\int sin^{2} (a x) d x

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

Solution steps

\int sin^{2} (a x) d x

Rewrite using trig identities

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

\int 1 d x = x

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

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

Simplify

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

= \frac{1}{2} x - \frac{1}{4 a} sin (2 a x)

Add a constant to the solution

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

\frac{\partial}{\partial x} (\frac{1}{x ^{2} - y ^{2}})

\frac{d y}{d x} = e^{5 x + 3 y}

y^{^{''}} + 4 y = 3 tan (2 x)

\int sin (2 x - 5) d x

\int_{0}^{2 π} 7 x^{2} sin (\frac{1}{4} x) d x

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