Practicing Success
$\int \frac{x}{\sqrt{1+x^2+\sqrt{\left(1+x^2\right)^3}}} d x$ is equal to |
$\frac{1}{2} \ln \left(1+\sqrt{1+x^2}\right)+C$ $\frac{-2}{3\left(1+\sqrt{1+x^2}\right)^{3 / 2}}+C$ $2\left(1+\sqrt{1+x^2}\right)+C$ $2 \sqrt{1+\sqrt{1+x^2}}+C$ |
$2 \sqrt{1+\sqrt{1+x^2}}+C$ |
Let $1+x^2=t^2$. Then, $x d x=t d t$ ∴ $I=\int \frac{x}{\sqrt{1+x^2+\sqrt{\left(1+x^2\right)^3}}} d x$ $\Rightarrow I=\int \frac{t}{\sqrt{t^2+t^3}} d t=\int \frac{1}{\sqrt{1+t}} d t=2 \sqrt{1+t}+C$ $\Rightarrow I=2 \sqrt{1+\sqrt{1+x^2}}+C$ |