What is the integral of (3x+4)/(sqrt(4-x^2)) ?

The integral of (3x+4)/(sqrt(4-x^2)) is -3sqrt(4-x^2)+4arcsin(1/2 x)+C

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integral of (3x+4)/(sqrt(4-x^2))

\int \frac{3 x + 4}{4 - x ^{2}} d x

- 3 4 - x^{2} + 4 arcsin (\frac{1}{2} x) + C

Solution steps

\int \frac{3 x + 4}{4 - x ^{2}} d x

Expand

\frac{3 x + 4}{4 - x ^{2}} : \frac{3 x}{4 - x ^{2}} + \frac{4}{4 - x ^{2}}

= \int \frac{3 x}{4 - x ^{2}} + \frac{4}{4 - 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{3 x}{4 - x ^{2}} d x + \int \frac{4}{4 - x ^{2}} d x

\int \frac{3 x}{4 - x ^{2}} d x = - 3 4 - x^{2}

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

= - 3 4 - x^{2} + 4 arcsin (\frac{1}{2} x)

Add a constant to the solution

= - 3 4 - x^{2} + 4 arcsin (\frac{1}{2} x) + C

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What is the integral of (3x+4)/(sqrt(4-x^2)) ?
The integral of (3x+4)/(sqrt(4-x^2)) is -3sqrt(4-x^2)+4arcsin(1/2 x)+C