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

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

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integral from 0 to pi/3 of (9x-2)cos(3x)

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

- 2

Solution steps

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

Expand

(9 x - 2) cos (3 x) : 9 x cos (3 x) - 2 cos (3 x)

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

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

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

= - 2 - 0

Simplify

= - 2

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\int_{5}^{\infty} x e^{- \frac{x}{2}} d x

\int_{1}^{2} x^{3} - 2 x^{2} - 5 x + 6 d x

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