Question

Question #a4188

Answer 1

We need to integrate `f(x)=2x+5/\sqrt(1-x^2)`

First of all, the integral of the sum is the sum of the integrals, so we can trade the two addends separately.

The antiderivative of `2x` is easily `2/3 x^3`, given the rule `\int x^n=\frac{x^{n+1}}{n+1}`

As for `\int 1/\sqrt(1-x^2)`, we know that `d/{dx} \sin^{-1}(x) = 1/\sqrt(1-x^2)`, where with `\sin^{-1}(x)` I mean the inverse sine function.

Summing up the two results, the antiderivative is `2/3 x^3+5\sin^{-1}(x)+c`