∫ln2ln3xsin x 2sin x 2+ sin (ln 6- x2 )  dx is ____.

is ____.

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Solution:

Let,
I=

Put ${x}^{2}=t$
$⇒$2xdx=dt
The integral will transform into
I=

Adding both the equations, we get
2I= $\frac{1}{2}{\int }_{\mathit{ln}2}^{\mathit{ln}3}\frac{\mathit{sint}+\mathit{sin}\left(\mathit{ln}6-t\right)}{\mathit{sint}+\mathit{sin}\left(\mathit{ln}6-t\right)}$ dt
$⇒$I= $\frac{1}{4}{\int }_{\mathit{ln}2}^{\mathit{ln}3}\mathit{dt}$
$⇒I=\frac{1}{4}\left[\mathit{ln}3-\mathit{ln}2\right]$
$⇒I=\frac{1}{4}\mathit{ln}\frac{3}{2}$
Hence option 1 is correct.

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