What is the integral from pi/4 to pi/2 of cos(3t) ?

The integral from pi/4 to pi/2 of cos(3t) is (-sqrt(2)-1)/(3sqrt(2))

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integral from pi/4 to pi/2 of cos(3t)

\int_{\frac{π}{4}}^{\frac{π}{2}} cos (3 t) d t

\frac{- 2 - 1}{3 2}

Decimal

- 0.56903 \dots

Solution steps

\int_{\frac{π}{4}}^{\frac{π}{2}} cos (3 t) d t

Apply u-substitution

: \int_{\frac{3 π}{4}}^{\frac{3 π}{2}} cos (u) \frac{1}{3} d u

= \int_{\frac{3 π}{4}}^{\frac{3 π}{2}} cos (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int_{\frac{3 π}{4}}^{\frac{3 π}{2}} cos (u) d u

Use the common integral:

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

= \frac{1}{3} [sin (u)]_{\frac{3 π}{4}}^{\frac{3 π}{2}}

Compute the boundaries:

- 1 - \frac{1}{2}

= \frac{1}{3} (- 1 - \frac{1}{2})

Simplify

= \frac{- 2 - 1}{3 2}

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What is the integral from pi/4 to pi/2 of cos(3t) ?
The integral from pi/4 to pi/2 of cos(3t) is (-sqrt(2)-1)/(3sqrt(2))