What is the integral of ((t+1)^3)/(t^2) ?

The integral of ((t+1)^3)/(t^2) is (t^2)/2+3t+3ln|t|-1/t+C

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integral of ((t+1)^3)/(t^2)

\int \frac{( t + 1 ) ^{3}}{t ^{2}} d t

\frac{t ^{2}}{2} + 3 t + 3 ln ∣ t ∣ - \frac{1}{t} + C

Solution steps

\int \frac{( t + 1 ) ^{3}}{t ^{2}} d t

Expand

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

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

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

\int 3 d t = 3 t

\int \frac{3}{t} d t = 3 ln ∣ t ∣

\int \frac{1}{t ^{2}} d t = - \frac{1}{t}

= \frac{t ^{2}}{2} + 3 t + 3 ln ∣ t ∣ - \frac{1}{t}

Add a constant to the solution

= \frac{t ^{2}}{2} + 3 t + 3 ln ∣ t ∣ - \frac{1}{t} + C

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What is the integral of ((t+1)^3)/(t^2) ?
The integral of ((t+1)^3)/(t^2) is (t^2)/2+3t+3ln|t|-1/t+C