Find the integral of the function 1x2+6x+5 14ln|x+1x+5|

Find the integral of the function $\frac{1}{x^{2} + 6 x + 5}$

$\frac{1}{4} l n | \frac{(x + 1)}{(x + 5)} | + c$

$\frac{1}{2} l n | \frac{(x + 3)}{(x - 3)} | + c$

$\frac{1}{2} l n | \frac{(x + 1)}{(x + 5)} | + c$

$\frac{1}{4} l n | \frac{(x + 3)}{(x - 3)} | + c$

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Right Answer is: A

SOLUTION

$\int \frac{d x}{x^{2} + 6 x + 5}$
When we get an integral of this form we try to express it as perfect square and proceed.
$\Rightarrow \int \frac{d x}{x^{2} + 6 x + 5} = \int \frac{d x}{(x + 3)^{2} - 4}$
This integral is of the form $\int \frac{d x}{x^{2} - a^{2}}$ which will be equal to $\frac{1}{2 a}$ ln $| \frac{x - a}{x + a} | + c$
$\Rightarrow \int \frac{d x}{(x + 3)^{2} - 4} = \frac{1}{4} l n | \frac{(x + 3) - 2}{(x + 3) + 2} | + c \frac{1}{4} l n | \frac{(x + 1)}{(x + 5)} | + c$

Find the integral of the function 1x2+6x+5 14ln|x+1x+5|

Right Answer is: A

SOLUTION

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