What is the integral of x/(-x-1) ?

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

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integral of x/(-x-1)

\int \frac{x}{- x - 1} d x

\frac{x ^{2}}{- x - 1} + ln ∣ x + 1 ∣ - \frac{1}{- x - 1} + C

Solution steps

\int \frac{x}{- x - 1} d x

Apply Integration By Parts

: \frac{x ^{2}}{- x - 1} - \int - \frac{x}{( - x - 1 ) ^{2}} d x

= \frac{x ^{2}}{- x - 1} - \int - \frac{x}{( - x - 1 ) ^{2}} d x

\int - \frac{x}{( - x - 1 ) ^{2}} d x = - ln ∣ x + 1 ∣ + \frac{1}{- x - 1}

= \frac{x ^{2}}{- x - 1} - (- ln ∣ x + 1 ∣ + \frac{1}{- x - 1})

Simplify

= \frac{x ^{2}}{- x - 1} + ln ∣ x + 1 ∣ - \frac{1}{- x - 1}

Add a constant to the solution

= \frac{x ^{2}}{- x - 1} + ln ∣ x + 1 ∣ - \frac{1}{- x - 1} + C

\int 4 x^{3} + 2 x y^{2} ln (x^{2} + y^{2}) d x

(x^{2} - x y) d x + x d y = 0

\int \frac{x ^{3} + 2}{x ^{2} + 8 x + 7} d x

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

n = 0 \sum \infty \frac{7}{5 ^{n}}

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