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

The integral of (x+2)/(x+3) is -ln|x+3|+x+C

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integral of (x+2)/(x+3)

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

- ln ∣ x + 3 ∣ + x + C

Solution steps

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

\frac{x + 2}{x + 3} = - \frac{1}{x + 3} + 1

= \int - \frac{1}{x + 3} + 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{1}{x + 3} d x + \int 1 d x

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

\int 1 d x = x

= - ln ∣ x + 3 ∣ + x

Add a constant to the solution

= - ln ∣ x + 3 ∣ + x + C

- 10 x^{2} y + y^{^{'}} = 9 x^{2}

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

y^{^{''}} - 6 y^{^{'}} + 9 y = 0, y (0) = - 3, y^{^{'}} (0) = 5

\frac{d y}{d x} = (x + 1)^{2}

\int_{1}^{3} (x + 2) (x - 1)^{5} d x

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