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

The integral of (a+x)/(sqrt(a^2-x^2)) is aarcsin(x/a)-sqrt(a^2-x^2)+C

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integral of (a+x)/(sqrt(a^2-x^2))

\int \frac{a + x}{a ^{2} - x ^{2}} d x

a arcsin (\frac{x}{a}) - a^{2} - x^{2} + C

Solution steps

\int \frac{a + x}{a ^{2} - x ^{2}} d x

Expand

\frac{a + x}{a ^{2} - x ^{2}} : \frac{a}{a ^{2} - x ^{2}} + \frac{x}{a ^{2} - x ^{2}}

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

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

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

= a arcsin (\frac{x}{a}) - a^{2} - x^{2}

Add a constant to the solution

= a arcsin (\frac{x}{a}) - a^{2} - x^{2} + C

What is the integral of (a+x)/(sqrt(a^2-x^2)) ?
The integral of (a+x)/(sqrt(a^2-x^2)) is aarcsin(x/a)-sqrt(a^2-x^2)+C