What is the integral of (x^2)/((x+1)) ?

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

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integral of (x^2)/((x+1))

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

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

Solution steps

\int \frac{x ^{2}}{x + 1} 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 + 1

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

Add a constant to the solution

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

\frac{d}{d x} (- e^{x} cos (x))

f (x) = 2 7 x + 1, at x = 9

1.5 e^{2 x}

f (x) = x \cdot sin (x)

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