What is the integral of (x+3)^{-1} ?

The integral of (x+3)^{-1} is x/(x+3)+ln|x+3|+3/(x+3)+C

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integral of (x+3)^{-1}

\int (x + 3)^{- 1} d x

\frac{x}{x + 3} + ln ∣ x + 3 ∣ + \frac{3}{x + 3} + C

Solution steps

\int (x + 3)^{- 1} d x

Apply Integration By Parts

: \frac{x}{x + 3} - \int - \frac{x}{( x + 3 ) ^{2}} d x

= \frac{x}{x + 3} - \int - \frac{x}{( x + 3 ) ^{2}} d x

\int - \frac{x}{( x + 3 ) ^{2}} d x = - ln ∣ x + 3 ∣ - \frac{3}{x + 3}

= \frac{x}{x + 3} - (- ln ∣ x + 3 ∣ - \frac{3}{x + 3})

Simplify

= \frac{x}{x + 3} + ln ∣ x + 3 ∣ + \frac{3}{x + 3}

Add a constant to the solution

= \frac{x}{x + 3} + ln ∣ x + 3 ∣ + \frac{3}{x + 3} + C

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What is the integral of (x+3)^{-1} ?
The integral of (x+3)^{-1} is x/(x+3)+ln|x+3|+3/(x+3)+C