What is the integral from 0 to pi of sin(3x)sin(5x) ?

The integral from 0 to pi of sin(3x)sin(5x) is 0

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integral from 0 to pi of sin(3x)sin(5x)

\int_{0}^{π} sin (3 x) sin (5 x) d x

0

Solution steps

\int_{0}^{π} sin (3 x) sin (5 x) d x

Rewrite using trig identities

= \int_{0}^{π} \frac{1}{2} (- cos (8 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 (8 x) + cos (- 2 x) d x

Simplify

- cos (8 x) + cos (- 2 x) : cos (2 x) - cos (8 x)

= \frac{1}{2} \cdot \int_{0}^{π} cos (2 x) - cos (8 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 (8 x) d x)

\int_{0}^{π} cos (2 x) d x = 0

\int_{0}^{π} cos (8 x) d x = 0

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

Simplify

= 0

\frac{d}{d x} (\frac{4 x ^{2} + 4 x + 6}{x})

\int_{3}^{6} (x^{2} - 2 x) d x

\int \frac{1}{( x ^{2} - 4 ) ^{2}} d x

f (x) = sin (tan (3 x))

- x^{2} + 2 x + 4, at x = 3

What is the integral from 0 to pi of sin(3x)sin(5x) ?
The integral from 0 to pi of sin(3x)sin(5x) is 0