What is the integral from-pi to pi of sin(2x)sin(3x) ?

The integral from-pi to pi of sin(2x)sin(3x) is 0

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integral from-pi to pi of sin(2x)sin(3x)

\int_{- π}^{π} sin (2 x) sin (3 x) d x

0

Solution steps

\int_{- π}^{π} sin (2 x) sin (3 x) d x

Rewrite using trig identities

= \int_{- π}^{π} \frac{1}{2} (- cos (5 x) + cos (- 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_{- π}^{π} - cos (5 x) + cos (- x) d x

Simplify

- cos (5 x) + cos (- x) : cos (x) - cos (5 x)

= \frac{1}{2} \cdot \int_{- π}^{π} cos (x) - 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_{- π}^{π} cos (x) d x - \int_{- π}^{π} cos (5 x) d x)

\int_{- π}^{π} cos (x) d x = 0

\int_{- π}^{π} cos (5 x) d x = 0

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

Simplify

= 0

f (x) = e^{x^{5}}

\int \frac{10}{10 + e ^{x}} d x

\frac{d}{d x} (\frac{( x + y )}{( x - y )})

x \to 0 lim (- 3 - 2 x)

\int (1 - x^{2})^{- \frac{3}{2}} d x

What is the integral from-pi to pi of sin(2x)sin(3x) ?
The integral from-pi to pi of sin(2x)sin(3x) is 0