The value of \int_{\sqrt{ln \textrm{ } 2}}^{\sqrt{ln \textrm{ } 3}} \frac{x \textrm{ } \left(sinx\right)^{2}}{\left(sinx\right)^{2} + sin \textrm{ } \left(\right. ln \textrm{ } 6

Engineering

Mathematics

Properties of Definite Integral

Question

The value of $\int_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x {sinx}^{2}}{{sinx}^{2} + \sin (\ln 6 - x^{2})} dx$ is :

$\ln \frac{3}{2}$

$\frac{1}{2} \ln \frac{3}{2}$

$\frac{1}{4} \ln \frac{3}{2}$

$\frac{1}{6} \ln \frac{3}{2}$

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$Let x^{2} = t xdx = \frac{dt}{2}$

$I = \frac{1}{2} \int_{ℓn 2}^{ℓn 3} \frac{sint}{sint + \sin (ℓn 6 - t)} dt$ ...............(1)

$I = \frac{1}{2} \int_{ℓn 2}^{ℓn 3} \frac{\sin (ℓn 6 - t)}{sint + \sin (ℓn 6 - t)} dt$ ...............(2)

Add (1) & (2)

$2 I = \frac{1}{2} \int_{ℓn 2}^{ℓn 3} dt$

$I = \frac{1}{4} (ℓn 3 - ℓn 2) = \frac{1}{4} ℓn \frac{3}{2}$