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Q.

$\int_{\ln 2}^{\ln 3} \frac{x {\sin x}^{2}}{{\sin x}^{2} + \sin (\ln 6 - x^{2})} dx$ is ____.

see full answer

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Detailed Solution

Let,
I=

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

Put

x^{2} = t

\Rightarrow

2xdx=dt
The integral will transform into
I=

\int_{\ln 2}^{\ln 3} \frac{\frac{1}{2 sint}}{{si n x}^{2} + \sin (\ln 6 - t)} dx

= \int_{\ln 2}^{\ln 3} \frac{\frac{1}{2 \sin (\ln 6 - t)}}{{\sin x}^{2} + \sin (\ln 6 - t)} dx

Adding both the equations, we get
2I=

\frac{1}{2} \int_{\ln 2}^{\ln 3} \frac{sint + \sin (\ln 6 - t)}{sint + \sin (\ln 6 - t)}

dt

\Rightarrow

I=

\frac{1}{4} \int_{\ln 2}^{\ln 3} dt

\Rightarrow I = \frac{1}{4} [\ln 3 - \ln 2]

\Rightarrow I = \frac{1}{4} \ln \frac{3}{2}

Hence option 1 is correct.

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