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

The integral of ((t+2)^2)/(t^3) is ln|t|-4/t-2/(t^2)+C

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

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

ln ∣ t ∣ - \frac{4}{t} - \frac{2}{t ^{2}} + C

Solution steps

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

Expand

\frac{( t + 2 ) ^{2}}{t ^{3}} : \frac{1}{t} + \frac{4}{t ^{2}} + \frac{4}{t ^{3}}

= \int \frac{1}{t} + \frac{4}{t ^{2}} + \frac{4}{t ^{3}} d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{t} d t + \int \frac{4}{t ^{2}} d t + \int \frac{4}{t ^{3}} d t

\int \frac{1}{t} d t = ln ∣ t ∣

\int \frac{4}{t ^{2}} d t = - \frac{4}{t}

\int \frac{4}{t ^{3}} d t = - \frac{2}{t ^{2}}

= ln ∣ t ∣ - \frac{4}{t} - \frac{2}{t ^{2}}

Add a constant to the solution

= ln ∣ t ∣ - \frac{4}{t} - \frac{2}{t ^{2}} + C

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What is the integral of ((t+2)^2)/(t^3) ?
The integral of ((t+2)^2)/(t^3) is ln|t|-4/t-2/(t^2)+C