What is the integral of sin^2((3pix)/(10)) ?

The integral of sin^2((3pix)/(10)) is 1/2 (x-5/(3pi)sin((3pix)/5))+C

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integral of sin^2((3pix)/(10))

\int sin^{2} (\frac{3 π x}{10}) d x

\frac{1}{2} (x - \frac{5}{3 π} sin (\frac{3 π x}{5})) + C

Solution steps

\int sin^{2} (\frac{3 π x}{10}) d x

Rewrite using trig identities

= \int \frac{1}{2} (1 - cos (\frac{3 π x}{5})) 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 (\frac{3 π x}{5}) 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 (\frac{3 π x}{5}) d x)

\int 1 d x = x

\int cos (\frac{3 π x}{5}) d x = \frac{5}{3 π} sin (\frac{3 π x}{5})

= \frac{1}{2} (x - \frac{5}{3 π} sin (\frac{3 π x}{5}))

Add a constant to the solution

= \frac{1}{2} (x - \frac{5}{3 π} sin (\frac{3 π x}{5})) + C

\int cos (5 x) \cdot cos (3 x) d x

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\int cos (t - u) d u

\int \frac{z}{( y ^{2} + z ^{2} ) ^{\frac{3}{2}}} d z

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

What is the integral of sin^2((3pix)/(10)) ?
The integral of sin^2((3pix)/(10)) is 1/2 (x-5/(3pi)sin((3pix)/5))+C