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

The integral of (ax^2+x)/(3x+1) is a/(54)(9x^2-6x+2ln|3x+1|-3)+1/9 (3x+1-ln|3x+1|)+C

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

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

\frac{a}{54} (9 x^{2} - 6 x + 2 ln ∣ 3 x + 1 ∣ - 3) + \frac{1}{9} (3 x + 1 - ln ∣ 3 x + 1 ∣) + C

Solution steps

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

Expand

\frac{a x ^{2} + x}{3 x + 1} : \frac{a x ^{2}}{3 x + 1} + \frac{x}{3 x + 1}

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

\int \frac{a x ^{2}}{3 x + 1} d x = \frac{a}{54} (9 x^{2} - 6 x + 2 ln ∣ 3 x + 1 ∣ - 3)

\int \frac{x}{3 x + 1} d x = \frac{1}{9} (3 x + 1 - ln ∣ 3 x + 1 ∣)

= \frac{a}{54} (9 x^{2} - 6 x + 2 ln ∣ 3 x + 1 ∣ - 3) + \frac{1}{9} (3 x + 1 - ln ∣ 3 x + 1 ∣)

Add a constant to the solution

= \frac{a}{54} (9 x^{2} - 6 x + 2 ln ∣ 3 x + 1 ∣ - 3) + \frac{1}{9} (3 x + 1 - ln ∣ 3 x + 1 ∣) + C

\int \frac{4}{1 + sin ( x ) + cos ( x )} d x

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\int x^{2} cos^{2} (2 x) d x

What is the integral of (ax^2+x)/(3x+1) ?
The integral of (ax^2+x)/(3x+1) is a/(54)(9x^2-6x+2ln|3x+1|-3)+1/9 (3x+1-ln|3x+1|)+C