What is the integral from 0 to pi/6 of sin(6x) ?

The integral from 0 to pi/6 of sin(6x) is 1/3

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integral from 0 to pi/6 of sin(6x)

\int_{0}^{\frac{π}{6}} sin (6 x) d x

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

\int_{0}^{\frac{π}{6}} sin (6 x) d x

Apply u-substitution

: \int_{0}^{π} sin (u) \frac{1}{6} d u

= \int_{0}^{π} sin (u) \frac{1}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int_{0}^{π} sin (u) d u

Use the common integral:

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

= \frac{1}{6} [- cos (u)]_{0}^{π}

Compute the boundaries:

2

= \frac{1}{6} \cdot 2

Simplify

= \frac{1}{3}

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What is the integral from 0 to pi/6 of sin(6x) ?
The integral from 0 to pi/6 of sin(6x) is 1/3