The value of _{{ 2}}^{{ 3}} {x x²}{ x²+ ( 6-x²)} dx is

Correct Answer is : 14ln⁡321/4 3/241ln23

Correct Answer is : 14ln⁡321/4 3/241ln23

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Manipal Entrance Test 2015 Math Paper

Question : 43 of 70

Marks: +1, -0

The value of

\int\limits_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x \sin x^{2}}{\sin x^{2}+\sin (\ln 6-x^{2})} dx

Solution:

Let

I = \int\limits_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x \sin x^{2}}{\sin x^{2}+\sin (\ln 6-x^{2})} dx

Putting

x^{2}=t

⇒

2x\,dx = dt

∴

I = \frac{1}{2} \int\limits_{\ln 2}^{\ln 3} \frac{\sin t}{\sin t+\sin (\ln 6-t)} dt

...(i)

Using

\int\limits_{a}^{b} f(x) dx = \int\limits_{a}^{b} f(a+b-x) dx

, we get

I = \frac{1}{2} \int\limits_{\ln 2}^{\ln 3} \frac{\sin (\ln 6-t)}{\sin (\ln 6-t)+\sin t} dt

....(ii)

On adding Eqs.(i) and (ii), we get

2 I = \int\limits_{\ln 2}^{\ln 3} 1\,dt = [t]_{\ln 2}^{\ln 3}

⇒

2 I = \frac{1}{2}(\ln 3-\ln 2) = \frac{1}{2} \ln \left( \frac{3}{2} \right)

⇒

I = \frac{1}{4} \ln \left( \frac{3}{2} \right)

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