What is the integral of (t^3+5t^2+7t+1)/(t^2+3t) ?

The integral of (t^3+5t^2+7t+1)/(t^2+3t) is (t^2)/2+2t+1/3 ln|t|+2/3 ln|t+3|+C

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integral of (t^3+5t^2+7t+1)/(t^2+3t)

\int \frac{t ^{3} + 5 t ^{2} + 7 t + 1}{t ^{2} + 3 t} d t

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

Solution steps

\int \frac{t ^{3} + 5 t ^{2} + 7 t + 1}{t ^{2} + 3 t} d t

Complete the square

t^{2} + 3 t : (t + \frac{3}{2})^{2} - \frac{9}{4}

= \int \frac{t ^{3} + 5 t ^{2} + 7 t + 1}{( t + \frac{3}{2} ) ^{2} - \frac{9}{4}} d t

Apply u-substitution

: \int \frac{20 u ^{2} - 32 u + 4 ( u - \frac{3}{2} ) ^{3} + 7}{4 u ^{2} - 9} d u

= \int \frac{20 u ^{2} - 32 u + 4 ( u - \frac{3}{2} ) ^{3} + 7}{4 u ^{2} - 9} d u

Apply u-substitution

: \int \frac{v ^{3} + 5 v ^{2} + 7 v + 1}{v ( v + 3 )} d v

= \int \frac{v ^{3} + 5 v ^{2} + 7 v + 1}{v ( v + 3 )} d v

Take the partial fraction of

\frac{v ^{3} + 5 v ^{2} + 7 v + 1}{v ( v + 3 )} : v + 2 + \frac{1}{3 v} + \frac{2}{3 ( v + 3 )}

= \int v + 2 + \frac{1}{3 v} + \frac{2}{3 ( v + 3 )} d v

Apply the Sum Rule:

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

= \int v d v + \int 2 d v + \int \frac{1}{3 v} d v + \int \frac{2}{3 ( v + 3 )} d v

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

\int 2 d v = 2 v

\int \frac{1}{3 v} d v = \frac{1}{3} ln ∣ v ∣

\int \frac{2}{3 ( v + 3 )} d v = \frac{2}{3} ln ∣ v + 3 ∣

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of (t^3+5t^2+7t+1)/(t^2+3t) ?
The integral of (t^3+5t^2+7t+1)/(t^2+3t) is (t^2)/2+2t+1/3 ln|t|+2/3 ln|t+3|+C