What is the integral from 2 to 3 of 4sin(pi/3 x) ?

The integral from 2 to 3 of 4sin(pi/3 x) is 6/pi

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integral from 2 to 3 of 4sin(pi/3 x)

\int_{2}^{3} 4 sin (\frac{π}{3} x) d x

\frac{6}{π}

Decimal Notation

1.90985 \dots

Solution steps

\int_{2}^{3} 4 sin (\frac{π}{3} x) d x

Take the constant out:

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

= 4 \cdot \int_{2}^{3} sin (\frac{π}{3} x) d x

Apply u-substitution

: \int_{\frac{2 π}{3}}^{π} sin (u) \frac{3}{π} d u

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

Take the constant out:

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

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

Use the common integral:

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

= 4 \cdot \frac{3}{π} [- cos (u)]_{\frac{2 π}{3}}^{π}

Simplify

4 \cdot \frac{3}{π} [- cos (u)]_{\frac{2 π}{3}}^{π} : \frac{12}{π} [- cos (u)]_{\frac{2 π}{3}}^{π}

= \frac{12}{π} [- cos (u)]_{\frac{2 π}{3}}^{π}

Compute the boundaries:

\frac{1}{2}

= \frac{12}{π} \cdot \frac{1}{2}

Simplify

= \frac{6}{π}

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What is the integral from 2 to 3 of 4sin(pi/3 x) ?
The integral from 2 to 3 of 4sin(pi/3 x) is 6/pi