Evaluate ∫ (2 x + 3)/(x^2 + 3 x + 5) dx.

(x (2 x + 3))/(x^2 + 3 x + 5) + C

Question 23

MediumOriginal question

Evaluate $\int \frac{2 x + 3}{x^{2} + 3 x + 5}\,dx$ .

2 \log\left|x^{2} + 3 x + 5\right| + C

Correct answerYour answer

\log\left|x^{2} + 3 x + 5\right| + C

Correct answerYour answer

- \log\left|x^{2} + 3 x + 5\right| + C

Correct answerYour answer

\frac{x \left(2 x + 3\right)}{x^{2} + 3 x + 5} + C

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

Put $t = x^2 + 3x + 5$ ; then $dt$ is the rest of the integrand, up to a constant.
The integral becomes a standard form in $t$ ; putting $t = x^2 + 3x + 5$ back gives $\log\left|x^{2} + 3 x + 5\right| + C$ .
Check: differentiating the answer gives back the integrand.

Worth noting

Getting the constant from $dt$ wrong multiplies or divides the answer: check by differentiating.

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