What is the integral from-1/2 to 1/2 of cos(pix) ?

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

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integral from-1/2 to 1/2 of cos(pix)

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

\frac{2}{π}

Decimal

0.63661 \dots

Solution steps

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

Apply u-substitution

: \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (u) \frac{1}{π} d u

= \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (u) \frac{1}{π} d u

Take the constant out:

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

= \frac{1}{π} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (u) d u

Use the common integral:

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

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

Compute the boundaries:

2

= \frac{1}{π} \cdot 2

Simplify

= \frac{2}{π}

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