Question
Let $f(x)=\int \frac{x^{2}}{\left(1+x^{2}\right)\left(1+\sqrt{1+x^{2}}\right)} d x$ and $f(0)=0$. Then$f(1)$ is equal to(a) $\log _{e}(1+\sqrt{2})$(b) $\log _{e}(1+\sqrt{2})-\frac{\pi}{4}$(c) $\log _{e}(1+\sqrt{2})+\frac{\pi}{4}$(d) None of these
Step 1
So, $I = \int \frac{x^{2}}{\left(1+x^{2}\right)\left(1+\sqrt{1+x^{2}}\right)} dx$. We can make the substitution $x = \tan \theta$. This implies $dx = \sec^{2} \theta d\theta$. Also, we know that $1 + \tan^{2} \theta = \sec^{2} \theta$. Show more…
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