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

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

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integral from 0 to pi of sin(x)sin(3x)

\int_{0}^{π} sin (x) sin (3 x) d x

0

Solution steps

\int_{0}^{π} sin (x) sin (3 x) d x

Rewrite using trig identities

= \int_{0}^{π} \frac{1}{2} (- cos (4 x) + cos (- 2 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_{0}^{π} - cos (4 x) + cos (- 2 x) d x

Simplify

- cos (4 x) + cos (- 2 x) : cos (2 x) - cos (4 x)

= \frac{1}{2} \cdot \int_{0}^{π} cos (2 x) - cos (4 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_{0}^{π} cos (2 x) d x - \int_{0}^{π} cos (4 x) d x)

\int_{0}^{π} cos (2 x) d x = 0

\int_{0}^{π} cos (4 x) d x = 0

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

Simplify

= 0

\frac{x ^{2}}{x ^{2} + 3}

y^{^{'}} = x^{2} y - y + x^{2} - 1

\int e [2 x] d x

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

\int_{0.1}^{1} 1.5 x^{2} d x

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