What is the integral of (5r^2)/(r+1) ?

The integral of (5r^2)/(r+1) is 5(((r+1)^2)/2-2(r+1)+ln|r+1|)+C

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integral of (5r^2)/(r+1)

\int \frac{5 r ^{2}}{r + 1} d r

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

Solution steps

\int \frac{5 r ^{2}}{r + 1} d r

Take the constant out:

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

= 5 \cdot \int \frac{r ^{2}}{r + 1} d r

Apply u-substitution

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

= 5 \cdot \int \frac{( u - 1 ) ^{2}}{u} d u

Expand

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

= 5 \cdot \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

= 5 (\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 ∣

= 5 (\frac{u ^{2}}{2} - 2 u + ln ∣ u ∣)

Substitute back

u = r + 1

= 5 (\frac{( r + 1 ) ^{2}}{2} - 2 (r + 1) + ln ∣ r + 1 ∣)

Add a constant to the solution

= 5 (\frac{( r + 1 ) ^{2}}{2} - 2 (r + 1) + ln ∣ r + 1 ∣) + C

f (x) = 9 x^{2} + 5

n = 0 \sum \infty \frac{1}{n ^{\frac{3}{2}}}

(x^{3} - 11 x^{2} + 30 x), (- x^{3} + 11 x^{2} - 30 x)

\frac{d}{d x} (lo g_{3} (\frac{x ^{2}}{x + 1}))

\frac{d y}{d x} = - 4 (y - 3)

What is the integral of (5r^2)/(r+1) ?
The integral of (5r^2)/(r+1) is 5(((r+1)^2)/2-2(r+1)+ln|r+1|)+C