What is the integral of (x+2)/(sqrt(x^2+9)) ?

The integral of (x+2)/(sqrt(x^2+9)) is sqrt(x^2+9)+2ln(1/3 |x+sqrt(9+x^2)|)+C

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integral of (x+2)/(sqrt(x^2+9))

\int \frac{x + 2}{x ^{2} + 9} d x

x^{2} + 9 + 2 ln (\frac{1}{3} x + 9 + x^{2}) + C

Solution steps

\int \frac{x + 2}{x ^{2} + 9} d x

Expand

\frac{x + 2}{x ^{2} + 9} : \frac{x}{x ^{2} + 9} + \frac{2}{x ^{2} + 9}

= \int \frac{x}{x ^{2} + 9} + \frac{2}{x ^{2} + 9} 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}{x ^{2} + 9} d x + \int \frac{2}{x ^{2} + 9} d x

\int \frac{x}{x ^{2} + 9} d x = x^{2} + 9

\int \frac{2}{x ^{2} + 9} d x = 2 ln (\frac{1}{3} x + 9 + x^{2})

= x^{2} + 9 + 2 ln (\frac{1}{3} x + 9 + x^{2})

Add a constant to the solution

= x^{2} + 9 + 2 ln (\frac{1}{3} x + 9 + x^{2}) + C

\int \frac{6}{x - 5} d x

\frac{\partial}{\partial y} (3 x^{2} y)

\int_{0}^{12} 1 + \frac{x}{4} d x

y = 14 (sin (x) + cos (x))

n = 0 \sum \infty e^{3 - 2 n}

What is the integral of (x+2)/(sqrt(x^2+9)) ?
The integral of (x+2)/(sqrt(x^2+9)) is sqrt(x^2+9)+2ln(1/3 |x+sqrt(9+x^2)|)+C