What is the integral of 10sin^2((5x)/2) ?

The integral of 10sin^2((5x)/2) is 5x-sin(5x)+C

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integral of 10sin^2((5x)/2)

\int 10 sin^{2} (\frac{5 x}{2}) d x

5 x - sin (5 x) + C

Solution steps

\int 10 sin^{2} (\frac{5 x}{2}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 10 \cdot \int sin^{2} (\frac{5 x}{2}) d x

Rewrite using trig identities

= 10 \cdot \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

= 10 \cdot \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

= 10 \cdot \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)

= 10 \cdot \frac{1}{2} (x - \frac{1}{5} sin (5 x))

Simplify

10 \cdot \frac{1}{2} (x - \frac{1}{5} sin (5 x)) : 5 x - sin (5 x)

= 5 x - sin (5 x)

Add a constant to the solution

= 5 x - sin (5 x) + C

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What is the integral of 10sin^2((5x)/2) ?
The integral of 10sin^2((5x)/2) is 5x-sin(5x)+C