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

The integral from-pi to pi of cos(x/2) is 4

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

\int_{- π}^{π} cos (\frac{x}{2}) d x

4

Solution steps

\int_{- π}^{π} cos (\frac{x}{2}) d x

Apply u-substitution

: \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (u) \cdot 2 d u

= \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (u) \cdot 2 d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= 2 [sin (u)]_{- \frac{π}{2}}^{\frac{π}{2}}

Compute the boundaries:

2

= 2 \cdot 2

Simplify

= 4

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