First slide

Methods of integration

Question

$Evaluate \int \frac{2 x}{(x^{2} + 1) (x^{2} + 2)} d x$

Easy

A

$\log |x^{2} + 1| - \log |x^{2} + 2| + C$
B

$\log |x^{2} + 2| - \log |x^{2} + 1| + C$
C

$\log |x^{2} + 1| + \log |x^{2} + 2| + C$
D

None of these

Solution

$Let I = \int \frac{2 x}{(x^{2} + 1) (x^{2} + 2)} d x$
$Putting x^{2} = t and 2 x d x = d t, we get$
$\begin{array}{l} I = \int \frac{d t}{(t + 1) (t + 2)} = \int (\frac{1}{t + 1} - \frac{1}{t + 2}) d t \\ = \log | t + 1 | - \log | t + 2 | + C \\ = \log |x^{2} + 1| - \log |x^{2} + 2| + C \end{array}$

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