First slide

Methods of integration

Question

$\int \frac{x^{2} - 1}{x^{4} + x^{2} + 1} dx$ is equal to

Moderate

A

$\log (x^{4} + x^{2} + 1) + C$
B

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

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

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

Solution

$\begin{array}{l} \int \frac{x^{2} - 1}{x^{4} + x^{2} + 1} dx \\ \Rightarrow I = \int \frac{1 - \frac{1}{x^{2}}}{x^{2} + \frac{1}{x^{2}} + 1} dx = \int \frac{1 - \frac{1}{x^{2}}}{{(x + \frac{1}{x})}^{2} - 1} dx \\ Put x + \frac{1}{x} = t \Rightarrow (1 - \frac{1}{x^{2}}) dx = dt \\ ∴ I = \int \frac{dt}{t^{2} - 1^{2}} = \frac{1}{2 \times 1} \log \frac{t - 1}{t + 1} + C \\ = \frac{1}{2} \log \frac{x^{2} + 1 - x}{x^{2} + 1 + x} + C \end{array}$

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