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

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

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

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

0

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

: \int_{- πn}^{0} \frac{cos ( u )}{n} d u

= \frac{π}{2} \cdot \int_{- πn}^{0} \frac{cos ( u )}{n} d u

Take the constant out:

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

= \frac{π}{2} \cdot \frac{1}{n} \cdot \int_{- πn}^{0} cos (u) d u

Use the common integral:

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

= \frac{π}{2} \cdot \frac{1}{n} [sin (u)]_{- πn}^{0}

Simplify

\frac{π}{2} \cdot \frac{1}{n} [sin (u)]_{- πn}^{0} : \frac{π}{2 n} [sin (u)]_{- πn}^{0}

= \frac{π}{2 n} [sin (u)]_{- πn}^{0}

Compute the boundaries:

0

= \frac{π}{2 n} \cdot 0

Simplify

= 0

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