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

The integral of 1/(4x^3+4x^2+x) is ln|x|-ln|2x+1|+1/(2x+1)+C

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

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

ln ∣ x ∣ - ln ∣ 2 x + 1 ∣ + \frac{1}{2 x + 1} + C

Solution steps

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

Take the partial fraction of

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

= \int \frac{1}{x} - \frac{2}{2 x + 1} - \frac{2}{( 2 x + 1 ) ^{2}} 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{1}{x} d x - \int \frac{2}{2 x + 1} d x - \int \frac{2}{( 2 x + 1 ) ^{2}} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{2}{2 x + 1} d x = ln ∣ 2 x + 1 ∣

\int \frac{2}{( 2 x + 1 ) ^{2}} d x = - \frac{1}{2 x + 1}

= ln ∣ x ∣ - ln ∣ 2 x + 1 ∣ - (- \frac{1}{2 x + 1})

Simplify

= ln ∣ x ∣ - ln ∣ 2 x + 1 ∣ + \frac{1}{2 x + 1}

Add a constant to the solution

= ln ∣ x ∣ - ln ∣ 2 x + 1 ∣ + \frac{1}{2 x + 1} + C

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What is the integral of 1/(4x^3+4x^2+x) ?
The integral of 1/(4x^3+4x^2+x) is ln|x|-ln|2x+1|+1/(2x+1)+C