What is the integral from pi/5 to pi/3 of sin(2x) ?

The integral from pi/5 to pi/3 of sin(2x) is (sqrt(2)+sqrt(3-\sqrt{5)})/(4sqrt(2))

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integral from pi/5 to pi/3 of sin(2x)

\int_{\frac{π}{5}}^{\frac{π}{3}} sin (2 x) d x

\frac{2 + 3 - 5}{4 2}

Decimal

0.40450 \dots

Solution steps

\int_{\frac{π}{5}}^{\frac{π}{3}} sin (2 x) d x

Apply u-substitution

: \int_{\frac{2 π}{5}}^{\frac{2 π}{3}} sin (u) \frac{1}{2} d u

= \int_{\frac{2 π}{5}}^{\frac{2 π}{3}} sin (u) \frac{1}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int_{\frac{2 π}{5}}^{\frac{2 π}{3}} sin (u) d u

Use the common integral:

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

= \frac{1}{2} [- cos (u)]_{\frac{2 π}{5}}^{\frac{2 π}{3}}

Compute the boundaries:

\frac{1}{2} + \frac{3 - 5}{2 2}

= \frac{1}{2} (\frac{1}{2} + \frac{3 - 5}{2 2})

Simplify

= \frac{2 + 3 - 5}{4 2}

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What is the integral from pi/5 to pi/3 of sin(2x) ?
The integral from pi/5 to pi/3 of sin(2x) is (sqrt(2)+sqrt(3-\sqrt{5)})/(4sqrt(2))