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

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

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

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

ln x^{2} + 3 x + 5 + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{u} d u

= \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = x^{2} + 3 x + 5

= ln x^{2} + 3 x + 5

Add a constant to the solution

= ln x^{2} + 3 x + 5 + C

\int \frac{x ^{3} - 4 x ^{2} + 5 x - 1}{x ^{2} - 2 x + 1} d x

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

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

\int \frac{2}{x ^{2}} \frac{1}{x} + 9 d x

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