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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{( u + 1 ) ^{2}}{u} d u

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

Expand

\frac{( u + 1 ) ^{2}}{u} : u + 2 + \frac{1}{u}

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

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

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{u ^{2}}{2} + 2 u + ln ∣ u ∣

Substitute back

u = x + 2

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

Add a constant to the solution

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

\frac{d}{d x} (tan (x) ln (x))

x \to 3 lim (4 x^{3} - 3 x^{2} + x - 5)

\frac{\partial}{\partial y} (4 x y - 2 x^{2} - y^{4})

- 8 sin (x)

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

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