What is the integral of (5y^2+6y)/(10y+3) ?

The integral of (5y^2+6y)/(10y+3) is (y^2)/4+9/20 y-27/200 ln|10y+3|+C

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integral of (5y^2+6y)/(10y+3)

\int \frac{5 y ^{2} + 6 y}{10 y + 3} d y

\frac{y ^{2}}{4} + \frac{9}{20} y - \frac{27}{200} ln ∣ 10 y + 3 ∣ + C

Solution steps

\int \frac{5 y ^{2} + 6 y}{10 y + 3} d y

Expand

\frac{5 y ^{2} + 6 y}{10 y + 3} : \frac{5 y ^{2}}{10 y + 3} + \frac{6 y}{10 y + 3}

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{5 y ^{2}}{10 y + 3} d y + \int \frac{6 y}{10 y + 3} d y

\int \frac{5 y ^{2}}{10 y + 3} d y = 5 (\frac{y ^{2}}{20} - \frac{3}{100} y + \frac{9}{1000} ln ∣ 10 y + 3 ∣)

\int \frac{6 y}{10 y + 3} d y = 6 (\frac{1}{10} y - \frac{3}{100} ln ∣ 10 y + 3 ∣)

= 5 (\frac{y ^{2}}{20} - \frac{3}{100} y + \frac{9}{1000} ln ∣ 10 y + 3 ∣) + 6 (\frac{1}{10} y - \frac{3}{100} ln ∣ 10 y + 3 ∣)

Simplify

5 (\frac{y ^{2}}{20} - \frac{3}{100} y + \frac{9}{1000} ln ∣ 10 y + 3 ∣) + 6 (\frac{1}{10} y - \frac{3}{100} ln ∣ 10 y + 3 ∣) : \frac{y ^{2}}{4} + \frac{9}{20} y - \frac{27}{200} ln ∣ 10 y + 3 ∣

= \frac{y ^{2}}{4} + \frac{9}{20} y - \frac{27}{200} ln ∣ 10 y + 3 ∣

Add a constant to the solution

= \frac{y ^{2}}{4} + \frac{9}{20} y - \frac{27}{200} ln ∣ 10 y + 3 ∣ + C

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What is the integral of (5y^2+6y)/(10y+3) ?
The integral of (5y^2+6y)/(10y+3) is (y^2)/4+9/20 y-27/200 ln|10y+3|+C