$\int \frac{2 x}{(x^{2} + 1) (x^{2} + 2)} dx is equal to$

log ∣ ∣ \frac{x^{2} + 1}{x^{2} + 2} ∣ ∣ + C

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log ∣ ∣ \frac{x^{2} - 1}{x^{2} + 2} ∣ ∣ + C

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log ∣ ∣ \frac{x^{2} + 1}{x^{2} - 2} ∣ ∣ + C

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Solution

The correct option is A $log ∣ ∣ \frac{x^{2} + 1}{x^{2} + 2} ∣ ∣ + C$
Here, if we make a substitution $t = x^{2}, we get dt = 2 xdx .$
Now, our integral becomes:
$\int \frac{dt}{(t + 1) (t + 2)}$ .
Now, the integrand is a proper fraction, so we can use integration by partial fraction.
Now, we can write the integrand as:
$\frac{1}{(t + 1) (t + 2)} = \frac{a}{t + 1} + \frac{b}{t + 2} \Rightarrow 1 = a (t + 2) + b (t + 1) comparing coefficient of x and constant terms, we get : a + b = 0 and 2 a + b = 1 solving these two questions, we get : a = 1 and b = - 1 . So, our integral becomes : I = \int (\frac{1}{t + 1} - \frac{1}{t + 2}) dt \Rightarrow I = ln (t + 1 | - ln (t + 2 | + c \Rightarrow I = ln (\frac{t + 1}{t + 2} ∣ ∣ + c Substuting back t = x^{2}, we get \Rightarrow I = ln (\frac{x^{2} + 1}{x^{2} + 2} ∣ ∣ + c$

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