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

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

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

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

\frac{π}{2} - 2

Decimal

- 0.42920 \dots

Solution steps

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

Apply Integration By Parts

: [sin (x) (x - 1) - \int sin (x) d x]_{0}^{\frac{π}{2}}

= [sin (x) (x - 1) - \int sin (x) d x]_{0}^{\frac{π}{2}}

\int sin (x) d x = - cos (x)

= [sin (x) (x - 1) - (- cos (x))]_{0}^{\frac{π}{2}}

Simplify

= [sin (x) (x - 1) + cos (x)]_{0}^{\frac{π}{2}}

Compute the boundaries:

\frac{π}{2} - 2

= \frac{π}{2} - 2

\int_{\frac{π}{3}}^{x} sin^{2} (t) d t

\int_{0}^{8} - 2 x^{2} + 16 x d x

\int_{1}^{7} - \frac{1}{7} x + 1 d x

\int_{3}^{4} (x^{2} + x - 12) d x

\int_{1}^{2} x + 6 d x

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