What is the integral from 0 to pi/4 of sin(3x) ?

The integral from 0 to pi/4 of sin(3x) is (1+sqrt(2))/(3sqrt(2))

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integral from 0 to pi/4 of sin(3x)

\int_{0}^{\frac{π}{4}} sin (3 x) d x

\frac{1 + 2}{3 2}

Decimal

0.56903 \dots

Solution steps

\int_{0}^{\frac{π}{4}} sin (3 x) d x

Apply u-substitution

= \int_{0}^{\frac{3 π}{4}} sin (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_{0}^{\frac{3 π}{4}} sin (u) d u

Use the common integral:

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

= \frac{1}{3} [- cos (u)]_{0}^{\frac{3 π}{4}}

Compute the boundaries:

\frac{1}{2} + 1

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

Simplify

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

What is the integral from 0 to pi/4 of sin(3x) ?
The integral from 0 to pi/4 of sin(3x) is (1+sqrt(2))/(3sqrt(2))