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NDA Math Definite Integrals

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For the next four (4) items that follow :

Consider the integral

I_m = \int\limits_{0}^{\pi} \frac{\sin 2mx}{\sin x} dx

where m is a positive integer.

[NDA I 2015]

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Question : 48 of 62

Marks: +1, -0

What is

I_1

0
1/2
1
2

Solution:

Substituting ‘m’ as 1, we get :

I_1 = \int\limits_{0}^{\pi} \frac{\sin 2x}{\sin x} dx

= \int\limits_{0}^{\pi} \frac{2 \sin x \cos x}{\sin x} dx

= \int\limits_{0}^{\pi} 2 \cos x dx

= 2[\sin x]^{\pi}_{0}

= 0

Hence for ‘m’ as 1, the integral yields 0.

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