What is the integral from 0 to pi of sin(7x)cos(3x) ?

The integral from 0 to pi of sin(7x)cos(3x) is 0

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integral from 0 to pi of sin(7x)cos(3x)

\int_{0}^{π} sin (7 x) cos (3 x) d x

0

Solution steps

\int_{0}^{π} sin (7 x) cos (3 x) d x

Use the following identity:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int_{0}^{π} \frac{sin ( 7 x + 3 x ) + sin ( 7 x - 3 x )}{2} 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} sin (7 x + 3 x) + sin (7 x - 3 x) d x

Simplify

= \frac{1}{2} \cdot \int_{0}^{π} sin (10 x) + sin (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} sin (10 x) d x + \int_{0}^{π} sin (4 x) d x)

\int_{0}^{π} sin (10 x) d x = 0

\int_{0}^{π} sin (4 x) d x = 0

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

Simplify

= 0

\int_{0}^{2 π} (1 - cos (x))^{3} d x

\int_{0}^{2 π} cos^{3} (t) d t

\int_{0}^{\frac{π}{2}} cos^{11} (x) d x

\int_{0}^{3} x^{6} d x

\int_{0}^{1} e^{2 x^{2}} d x

What is the integral from 0 to pi of sin(7x)cos(3x) ?
The integral from 0 to pi of sin(7x)cos(3x) is 0