"Evaluate the integrals. ∫0π201−cos⁡20d0" was answered by our community

Cem Hayes

Answered question

2021-10-18

Evaluate the integrals.

\int_{0}^{\frac{π}{2}} 0 \sqrt{1 - \cos 20} d 0

comentezq

Skilled2021-10-19Added 106 answers

Step 1
We have to evaluate the integral

\int_{0}^{\frac{π}{2}} \sqrt{1 - \cos 20} d 0

We know that,

1 - \cos 20 = 2 \sin^{2} 0

Therefore,

\int_{0}^{\frac{π}{2}} \sqrt{1 - \cos 20} d 0 = \int_{0}^{\frac{π}{2}} \sqrt{2 \sin^{2} 0} d 0

= \int_{0}^{\frac{π}{2}} \sqrt{2} \cdot \sin 0 d 0

= \sqrt{2} \int_{0}^{\frac{π}{2}} \sin 0 d 0

= \sqrt{2} {[- \cos 0]}_{0}^{\frac{π}{2}}

= \sqrt{2} [- \cos \frac{π}{2} + \cos 0]

= \sqrt{2} [- 0 + 1]

= \sqrt{2}

Step 2
Hence, required answer is

\sqrt{2}

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