Q. 315.0( 1 Vote )

Evaluate the foll

Answer :

Let


In the denominator, we have sin 2x = 2 sin x cos x


Note that we can write 2 sin x cos x = 1 – (1 – 2 sin x cos x)


We also have sin2x + cos2x = 1


1 – 2 sin x cos x = sin2x + cos2x – 2 sin x cos x


sin 2x = 1 – (sin x – cos x)2


So, using this, we can write our integral as




Now, put sin x – cos x = t


(cos x + sin x) dx = dt (Differentiating both sides)


When x = 0, t = sin 0 – cos 0 = 0 – 1 = -1


When,



So, the new limits are -1 and 0.


Substituting this in the original integral,




Recall,








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