What is the integral from 0 to L of sin^2((3pix)/L) ?

The integral from 0 to L of sin^2((3pix)/L) is 1/2 L

Popular >

integral from 0 to L of sin^2((3pix)/L)

\int_{0}^{L} sin^{2} (\frac{3 π x}{L}) d x

\frac{1}{2} L

Solution steps

\int_{0}^{L} sin^{2} (\frac{3 π x}{L}) d x

Rewrite using trig identities

= \int_{0}^{L} \frac{1}{2} (1 - cos (\frac{6 π x}{L})) 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_{0}^{L} 1 - cos (\frac{6 π x}{L}) 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_{0}^{L} 1 d x - \int_{0}^{L} cos (\frac{6 π x}{L}) d x)

\int_{0}^{L} 1 d x = L

\int_{0}^{L} cos (\frac{6 π x}{L}) d x = 0

= \frac{1}{2} (L - 0)

Simplify

= \frac{1}{2} L

\int_{1}^{4} (x^{2} - x - 6) d x

\int_{6.3}^{6.6} 12 x^{2} - 6 x + \frac{4}{x} d x

\int_{- 4}^{- 3} \frac{x}{( x ^{2} + 5 ) ^{2}} d x

\int_{0}^{4} (x^{2} - 1) d x

\int_{0}^{2} 12 x^{3} d x

What is the integral from 0 to L of sin^2((3pix)/L) ?
The integral from 0 to L of sin^2((3pix)/L) is 1/2 L