What is the integral from 0 to pi of (2x-1)cos(3x) ?

The integral from 0 to pi of (2x-1)cos(3x) is -4/9

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integral from 0 to pi of (2x-1)cos(3x)

\int_{0}^{π} (2 x - 1) cos (3 x) d x

- \frac{4}{9}

Decimal

- 0.44444 \dots

Solution steps

\int_{0}^{π} (2 x - 1) cos (3 x) d x

Expand

(2 x - 1) cos (3 x) : 2 x cos (3 x) - cos (3 x)

= \int_{0}^{π} 2 x cos (3 x) - cos (3 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

= \int_{0}^{π} 2 x cos (3 x) d x - \int_{0}^{π} cos (3 x) d x

\int_{0}^{π} 2 x cos (3 x) d x = - \frac{4}{9}

\int_{0}^{π} cos (3 x) d x = 0

= - \frac{4}{9} - 0

Simplify

= - \frac{4}{9}

What is the integral from 0 to pi of (2x-1)cos(3x) ?
The integral from 0 to pi of (2x-1)cos(3x) is -4/9