What is the integral of (x^2+5x+7)/(x^2+3x+4) ?

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

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integral of (x^2+5x+7)/(x^2+3x+4)

\int \frac{x ^{2} + 5 x + 7}{x ^{2} + 3 x + 4} d x

ln x^{2} + 3 x + 4 + x + C

Solution steps

\int \frac{x ^{2} + 5 x + 7}{x ^{2} + 3 x + 4} d x

\frac{x ^{2} + 5 x + 7}{x ^{2} + 3 x + 4} = \frac{2 x + 3}{x ^{2} + 3 x + 4} + 1

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

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

\int 1 d x = x

= ln x^{2} + 3 x + 4 + x

Add a constant to the solution

= ln x^{2} + 3 x + 4 + x + C

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