Resolved: "Evaluate the integral. ∫π6π2sin⁡2x2sin⁡x"

ZIHLOLEp3

Answered question

2021-11-09

Evaluate the integral.

\int_{\frac{π}{6}}^{\frac{π}{2}} \frac{\sin 2 x}{2 \sin x}

Mary Darby

Beginner2021-11-10Added 11 answers

Step 1
Given integral:

\int_{\frac{π}{6}}^{\frac{π}{2}} \frac{\sin 2 x}{2 \sin x}

Step 2
Integrating using the trigonometric identity:

\sin (2 x) = 2 \sin (x) \cos (x)

\int_{\frac{π}{6}}^{\frac{π}{2}} \frac{\sin 2 x}{2 \sin x} = \int_{\frac{π}{6}}^{\frac{π}{2}} \frac{2 \sin (x) \cos (x)}{2 \sin (x)} d x

= \int_{\frac{π}{6}}^{\frac{π}{2}} \cos (x) d x

Use the common integral:

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

= {[\sin (x)]}_{\frac{π}{6}}^{\frac{π}{2}}

= [\sin (\frac{π}{2}) - \sin (\frac{π}{6})]

= 1 - \frac{1}{2}

= \frac{1}{2}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?