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

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

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

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

π - 3

Decimal

0.14159 \dots

Solution steps

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

Expand

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

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

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

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

= π - 2 - 1

Simplify

= π - 3

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