# How do solve the following integral? int(t+7)^2/t^3dt

## $\int {\left(t + 7\right)}^{2} / {t}^{3} \mathrm{dt}$

Dec 4, 2017

Just expand the function and integrate. The answer is $\ln t - \frac{14}{t} - \frac{49}{2 {t}^{2}} + C$.

#### Explanation:

Let $f \left(t\right) = {\left(t + 7\right)}^{2} / {t}^{3}$. First, let's rearrange $f \left(t\right)$ into simpler terms.
$f \left(t\right) = \frac{{t}^{2} + 14 t + 49}{t} ^ 3$
$= \frac{1}{t} + \frac{14}{t} ^ 2 + \frac{49}{t} ^ 3$
$= {t}^{-} 1 + 14 {t}^{-} 2 + 49 {t}^{-} 3$

Then integrate $f \left(t\right)$. Note that $\int {x}^{n} \mathrm{dx} = \frac{1}{n + 1} {x}^{n + 1} + C$ if $n \ne - 1$, where $C$ is the integral constant. If $n = - 1$, $\int {x}^{-} 1 \mathrm{dx} = \int \left(\frac{1}{x}\right) \mathrm{dx} = \ln x + C$ is applied.

$\int f \left(t\right) \mathrm{dt} = \int \left({t}^{-} 1 + 14 {t}^{-} 2 + 49 {t}^{-} 3\right) \mathrm{dt}$
$= \int {t}^{-} 1 \mathrm{dt} + 14 \int {t}^{-} 2 \mathrm{dt} + 49 \int {t}^{-} 3 \mathrm{dt}$
$= \ln t + 14 \left(- {t}^{-} 1\right) + 49 \left(- \frac{1}{2} {t}^{-} 2\right) + C$
$= \ln t - \frac{14}{t} - \frac{49}{2 {t}^{2}} + C$.

Dec 4, 2017

$\ln \left(t\right) - \frac{7 \left(4 t + 7\right)}{2 {t}^{2}} + c$

#### Explanation:

solving ${\left(t + 7\right)}^{2}$

${\left(t + 7\right)}^{2} = {t}^{2} + 14 t + 49$

now in the integral

$\int \frac{{t}^{2} + 14 t + 49}{t} ^ 3 \mathrm{dt}$

separating the fractions

$\int \left({t}^{2} / {t}^{3} + 14 \frac{t}{t} ^ 3 + \frac{49}{t} ^ 3\right) \mathrm{dt}$

$\int \frac{1}{t} + \frac{14}{t} ^ 2 + \frac{49}{t} ^ 3 \mathrm{dt}$

separting

$\int \frac{1}{t} \mathrm{dt} = \ln \left(t\right) + c$

$\int \frac{14}{t} ^ 2 \mathrm{dt} = - \frac{14}{t} + c$

$\int \frac{49}{t} ^ 3 \mathrm{dt} = - \frac{49}{2 {t}^{2}} + c$

$\int \frac{1}{t} + \frac{14}{t} ^ 2 + \frac{49}{t} ^ 3 \mathrm{dt} = \ln \left(t\right) - \frac{14}{t} - \frac{49}{2 {t}^{2}} + c$

operating

$\int \frac{1}{t} + \frac{14}{t} ^ 2 + \frac{49}{t} ^ 3 \mathrm{dt} = \ln \left(t\right) - \frac{7 \left(4 t + 7\right)}{2 {t}^{2}} + c$

$\int \frac{{t}^{2} + 14 t + 49}{t} ^ 3 \mathrm{dt} = \ln \left(t\right) - \frac{7 \left(4 t + 7\right)}{2 {t}^{2}} + c$