What is the integral of (2x+3)/(sqrt(x^2+1)) ?

The integral of (2x+3)/(sqrt(x^2+1)) is 2sqrt(x^2+1)+3ln|sqrt(x^2+1)+x|+C

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integral of (2x+3)/(sqrt(x^2+1))

\int \frac{2 x + 3}{x ^{2} + 1} d x

2 x^{2} + 1 + 3 ln x^{2} + 1 + x + C

Solution steps

\int \frac{2 x + 3}{x ^{2} + 1} d x

Expand

\frac{2 x + 3}{x ^{2} + 1} : \frac{2 x}{x ^{2} + 1} + \frac{3}{x ^{2} + 1}

= \int \frac{2 x}{x ^{2} + 1} + \frac{3}{x ^{2} + 1} 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{2 x}{x ^{2} + 1} d x + \int \frac{3}{x ^{2} + 1} d x

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

\int \frac{3}{x ^{2} + 1} d x = 3 ln x^{2} + 1 + x

= 2 x^{2} + 1 + 3 ln x^{2} + 1 + x

Add a constant to the solution

= 2 x^{2} + 1 + 3 ln x^{2} + 1 + x + C

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What is the integral of (2x+3)/(sqrt(x^2+1)) ?
The integral of (2x+3)/(sqrt(x^2+1)) is 2sqrt(x^2+1)+3ln|sqrt(x^2+1)+x|+C