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Integral of $$$\frac{1}{\sqrt{x^{2} + 1}}$$$
Related calculator: Definite and Improper Integral Calculator
Your Input
Find $$$\int \frac{1}{\sqrt{x^{2} + 1}}\, dx$$$.
Solution
The integral of $$$\frac{1}{\sqrt{x^{2} + 1}}$$$ is $$$\int{\frac{1}{\sqrt{x^{2} + 1}} d x} = \operatorname{asinh}{\left(x \right)}$$$:
$${\color{red}{\int{\frac{1}{\sqrt{x^{2} + 1}} d x}}} = {\color{red}{\operatorname{asinh}{\left(x \right)}}}$$
Therefore,
$$\int{\frac{1}{\sqrt{x^{2} + 1}} d x} = \operatorname{asinh}{\left(x \right)}$$
Add the constant of integration:
$$\int{\frac{1}{\sqrt{x^{2} + 1}} d x} = \operatorname{asinh}{\left(x \right)}+C$$
Answer
$$$\int \frac{1}{\sqrt{x^{2} + 1}}\, dx = \operatorname{asinh}{\left(x \right)} + C$$$A