What is the integral of (x^2+4x)/(x+2) ?

The integral of (x^2+4x)/(x+2) is ((x+2)^2)/2-4ln|x+2|+C

Popular >

integral of (x^2+4x)/(x+2)

\int \frac{x ^{2} + 4 x}{x + 2} d x

\frac{( x + 2 ) ^{2}}{2} - 4 ln ∣ x + 2 ∣ + C

Solution steps

\int \frac{x ^{2} + 4 x}{x + 2} d x

Apply u-substitution

: \int \frac{u ^{2} - 4}{u} d u

= \int \frac{u ^{2} - 4}{u} d u

Expand

\frac{u ^{2} - 4}{u} : u - \frac{4}{u}

= \int u - \frac{4}{u} 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

= \int u d u - \int \frac{4}{u} d u

\int u d u = \frac{u ^{2}}{2}

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= \frac{u ^{2}}{2} - 4 ln ∣ u ∣

Substitute back

u = x + 2

= \frac{( x + 2 ) ^{2}}{2} - 4 ln ∣ x + 2 ∣

Add a constant to the solution

= \frac{( x + 2 ) ^{2}}{2} - 4 ln ∣ x + 2 ∣ + C

\int \frac{x ^{3}}{9 x ^{2} + 1} d x

\int \frac{1}{23 - x ^{2}} d x

\int \frac{7 x}{x + 3} d x

\int x^{2} (x^{3} + 1)^{100} d x

\int (x^{4} - 3 x^{\frac{3}{2}} + (\frac{7}{x}) + 5) d x

What is the integral of (x^2+4x)/(x+2) ?
The integral of (x^2+4x)/(x+2) is ((x+2)^2)/2-4ln|x+2|+C