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

The integral from 1/2 to 1/3 of cos(pi)x is 5/72

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integral from 1/2 to 1/3 of cos(pi)x

\int_{\frac{1}{2}}^{\frac{1}{3}} cos (π) x d x

\frac{5}{72}

Decimal

0.06944 \dots

Solution steps

\int_{\frac{1}{2}}^{\frac{1}{3}} cos (π) x d x

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{3}}^{\frac{1}{2}} cos (π) x d x

Take the constant out:

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

= - cos (π) \cdot \int_{\frac{1}{3}}^{\frac{1}{2}} x d x

cos (π) = - 1

= - (- 1) \cdot \int_{\frac{1}{3}}^{\frac{1}{2}} x d x

Apply the Power Rule

: [\frac{x ^{2}}{2}]_{\frac{1}{3}}^{\frac{1}{2}}

= - (- 1) [\frac{x ^{2}}{2}]_{\frac{1}{3}}^{\frac{1}{2}}

Simplify

= [\frac{x ^{2}}{2}]_{\frac{1}{3}}^{\frac{1}{2}}

Compute the boundaries:

\frac{5}{72}

= \frac{5}{72}

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What is the integral from 1/2 to 1/3 of cos(pi)x ?
The integral from 1/2 to 1/3 of cos(pi)x is 5/72