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

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

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

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

1 + x^{2} + ln x^{2} + 1 + x + C

Solution steps

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

Expand

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

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

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

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

= 1 + x^{2} + ln x^{2} + 1 + x

Add a constant to the solution

= 1 + x^{2} + ln x^{2} + 1 + x + C

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