What is the integral of (3(x+1)+4)/(x(2x+7)+3) ?

The integral of (3(x+1)+4)/(x(2x+7)+3) is 11/10 ln|2x+1|+2/5 ln|x+3|+C

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integral of (3(x+1)+4)/(x(2x+7)+3)

\int \frac{3 ( x + 1 ) + 4}{x ( 2 x + 7 ) + 3} d x

\frac{11}{10} ln ∣ 2 x + 1 ∣ + \frac{2}{5} ln ∣ x + 3 ∣ + C

Solution steps

\int \frac{3 ( x + 1 ) + 4}{x ( 2 x + 7 ) + 3} d x

Take the partial fraction of

\frac{3 ( x + 1 ) + 4}{x ( 2 x + 7 ) + 3} : \frac{11}{5 ( 2 x + 1 )} + \frac{2}{5 ( x + 3 )}

= \int \frac{11}{5 ( 2 x + 1 )} + \frac{2}{5 ( x + 3 )} d x

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{11}{5 ( 2 x + 1 )} d x + \int \frac{2}{5 ( x + 3 )} d x

\int \frac{11}{5 ( 2 x + 1 )} d x = \frac{11}{10} ln ∣ 2 x + 1 ∣

\int \frac{2}{5 ( x + 3 )} d x = \frac{2}{5} ln ∣ x + 3 ∣

= \frac{11}{10} ln ∣ 2 x + 1 ∣ + \frac{2}{5} ln ∣ x + 3 ∣

Add a constant to the solution

= \frac{11}{10} ln ∣ 2 x + 1 ∣ + \frac{2}{5} ln ∣ x + 3 ∣ + C

x \to 0 lim (\frac{( x ^{2} - 3 x )}{x})

y^{^{'}} - 5 y x^{4} = 0

\int a^{2}

\int (x^{2} sin (x^{3})) d x

\int \frac{x ^{3} - 1}{x + 2} d x

What is the integral of (3(x+1)+4)/(x(2x+7)+3) ?
The integral of (3(x+1)+4)/(x(2x+7)+3) is 11/10 ln|2x+1|+2/5 ln|x+3|+C