What is the integral of sin^2((5x)/2) ?

The integral of sin^2((5x)/2) is 1/2 (x-1/5 sin(5x))+C

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integral of sin^2((5x)/2)

\int sin^{2} (\frac{5 x}{2}) d x

\frac{1}{2} (x - \frac{1}{5} sin (5 x)) + C

Solution steps

\int sin^{2} (\frac{5 x}{2}) d x

Rewrite using trig identities

= \int \frac{1}{2} (1 - cos (5 x)) 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 (5 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 (5 x) d x)

\int 1 d x = x

\int cos (5 x) d x = \frac{1}{5} sin (5 x)

= \frac{1}{2} (x - \frac{1}{5} sin (5 x))

Add a constant to the solution

= \frac{1}{2} (x - \frac{1}{5} sin (5 x)) + C

(2 x + 1) y^{^{'}} + 2 y = 4 x

\frac{d}{d x} (1 - sec^{2} (x))

\int \frac{1}{5 y} d y

y^{^{''}} + y = 0, y (\frac{π}{3}) = 0, y^{^{'}} (\frac{π}{3}) = 8

\frac{\partial}{\partial x} (3 x + 9 x^{17})

What is the integral of sin^2((5x)/2) ?
The integral of sin^2((5x)/2) is 1/2 (x-1/5 sin(5x))+C