What is the integral of (3r^2)/(r+5) ?

The integral of (3r^2)/(r+5) is 3(((r+5)^2)/2-10(r+5)+25ln|r+5|)+C

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integral of (3r^2)/(r+5)

\int \frac{3 r ^{2}}{r + 5} d r

3 (\frac{( r + 5 ) ^{2}}{2} - 10 (r + 5) + 25 ln ∣ r + 5 ∣) + C

Solution steps

\int \frac{3 r ^{2}}{r + 5} d r

Take the constant out:

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

= 3 \cdot \int \frac{r ^{2}}{r + 5} d r

Apply u-substitution

: \int \frac{( u - 5 ) ^{2}}{u} d u

= 3 \cdot \int \frac{( u - 5 ) ^{2}}{u} d u

Expand

\frac{( u - 5 ) ^{2}}{u} : u - 10 + \frac{25}{u}

= 3 \cdot \int u - 10 + \frac{25}{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

= 3 (\int u d u - \int 10 d u + \int \frac{25}{u} d u)

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

\int 10 d u = 10 u

\int \frac{25}{u} d u = 25 ln ∣ u ∣

= 3 (\frac{u ^{2}}{2} - 10 u + 25 ln ∣ u ∣)

Substitute back

u = r + 5

= 3 (\frac{( r + 5 ) ^{2}}{2} - 10 (r + 5) + 25 ln ∣ r + 5 ∣)

Add a constant to the solution

= 3 (\frac{( r + 5 ) ^{2}}{2} - 10 (r + 5) + 25 ln ∣ r + 5 ∣) + C

(3, 5), (1, - 7)

x \to 0 lim (\frac{( 2 + x ^{3} ) - 8}{x})

\int_{0}^{2} \frac{5 t}{( t - 3 ) ^{2}} d t

\frac{d}{d x} (ln (- 5 x))

\frac{\frac{1}{3}}{s}

What is the integral of (3r^2)/(r+5) ?
The integral of (3r^2)/(r+5) is 3(((r+5)^2)/2-10(r+5)+25ln|r+5|)+C