What is the integral of (x+4)/(sqrt(1-x^2)) ?

The integral of (x+4)/(sqrt(1-x^2)) is -sqrt(1-x^2)+4arcsin(x)+C

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integral of (x+4)/(sqrt(1-x^2))

\int \frac{x + 4}{1 - x ^{2}} d x

- 1 - x^{2} + 4 arcsin (x) + C

Solution steps

\int \frac{x + 4}{1 - x ^{2}} d x

Expand

\frac{x + 4}{1 - x ^{2}} : \frac{x}{1 - x ^{2}} + \frac{4}{1 - x ^{2}}

= \int \frac{x}{1 - x ^{2}} + \frac{4}{1 - x ^{2}} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{1 - x ^{2}} d x + \int \frac{4}{1 - x ^{2}} d x

\int \frac{x}{1 - x ^{2}} d x = - 1 - x^{2}

\int \frac{4}{1 - x ^{2}} d x = 4 arcsin (x)

= - 1 - x^{2} + 4 arcsin (x)

Add a constant to the solution

= - 1 - x^{2} + 4 arcsin (x) + C

\int 1 + x^{- 2} d x

\int x^{2} \frac{1}{x + 1} d x

\int \frac{x y - 1}{x} d x

\int \frac{1}{( x ^{2} x ^{2} - 1 )} d x

What is the integral of (x+4)/(sqrt(1-x^2)) ?
The integral of (x+4)/(sqrt(1-x^2)) is -sqrt(1-x^2)+4arcsin(x)+C