What is the integral of (-2)/((u^2+u)) ?

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

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integral of (-2)/((u^2+u))

integral

\frac{- 2}{( u ^{2} + u )}

- 2 (ln ∣ u ∣ - ln ∣ u + 1 ∣) + C

Solution steps

\int \frac{- 2}{u ^{2} + u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 2 \cdot \int \frac{1}{u ^{2} + u} d u

Take the partial fraction of

\frac{1}{u ^{2} + u} : \frac{1}{u} - \frac{1}{u + 1}

= - 2 \cdot \int \frac{1}{u} - \frac{1}{u + 1} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - 2 (\int \frac{1}{u} d u - \int \frac{1}{u + 1} d u)

\int \frac{1}{u} d u = ln ∣ u ∣

\int \frac{1}{u + 1} d u = ln ∣ u + 1 ∣

= - 2 (ln ∣ u ∣ - ln ∣ u + 1 ∣)

Add a constant to the solution

= - 2 (ln ∣ u ∣ - ln ∣ u + 1 ∣) + C

5 x^{4} - 6 x^{2} + 3

\frac{22}{( 121 x ^{2} + 1 ) ^{2}}

\frac{2 ^{x}}{2 ^{x} ( 1 + 4 ^{x} )}

\frac{e ^{- x}}{x ^{\frac{1}{2}}}

cos^{2} (x) sin^{2} (x)

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