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

The integral of (x+1)/(sqrt(4+x^2)) is sqrt(4+x^2)+ln(1/2 |x+sqrt(4+x^2)|)+C

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

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

4 + x^{2} + ln (\frac{1}{2} x + 4 + x^{2}) + C

Solution steps

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

Expand

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

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

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

\int \frac{1}{4 + x ^{2}} d x = ln (\frac{1}{2} x + 4 + x^{2})

= 4 + x^{2} + ln (\frac{1}{2} x + 4 + x^{2})

Add a constant to the solution

= 4 + x^{2} + ln (\frac{1}{2} x + 4 + x^{2}) + C

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