Question

Evaluate:$\int \frac{dx}{(x^2+4x+13)}$

• A. $log(x^2+4x+3)+c$
• B. $\left(\frac{1}{3}\right){\tan }^{-1}\left(\frac{(x+2)}{3}\right)+c$
• C. $log(2x+4)+c$
• D. $\frac{[2x+4]}{{[x^2+4x+13]}^2}+c$

Answer 1

The correct option is B

$\left(\frac{1}{3}\right){\tan }^{-1}\left(\frac{(x+2)}{3}\right)+c$

Explanation for correct option:

Evaluating the integral

$$\begin{gathered} \int \frac{dx}{x^2-4x+13} \\ =\int \frac{dx}{x^2-2\times 2\times x+2^2-4+13} \\ =\int \frac{dx}{{(x-2)}^2-4+13}\left[\because a^2-2\times a\times b+b^2={(a-b)}^2\right] \\ =\int \frac{dx}{{(x-2)}^2+9} \\ =\int \frac{dx}{{(x-2)}^2+{\left(3\right)}^2} \\ =\frac{1}{3}{\tan }^{-1}\left(\frac{(x-2)}{3}\right)+c \end{gathered}$$

Hence, option (B) is correct.