Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{1}{\sqrt{\left(1-x^{2}\right)\left\{9+\left(\sin ^{-1} x\right)^{2}\right.}} d x$

Solution:

Let $\sin ^{-1} x=t$

$\mathrm{dt}=\frac{1}{\sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}$

Therefore, $\int \frac{1}{\sqrt{\left(1-x^{2}\right)\left\{9+\left(\sin ^{-1} x\right)^{2}\right.}} d x=\int \frac{1}{\sqrt{3^{2}-t^{2}}} d t$

Since we have, $\int \frac{1}{\sqrt{\left(x^{2}+a^{2}\right)}} d x=\log \left[x+\sqrt{\left.\left(x^{2}+a^{2}\right)\right]+c}\right.$

$=\int \frac{1}{\sqrt{3^{2}-t^{2}}} d t=\log \left[t+\sqrt{9+t^{2}}\right]+c$

$=\log \left[t+\sqrt{9+t^{2}}\right]+c=\log \left[\sin ^{-1} x+\sqrt{9+\left(\sin ^{-1} x\right)^{2}}\right]+c$

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