First slide

Methods of integration

Question

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

Moderate

A

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

$\frac{{xe}^{x}}{1 + x^{2}} + C$
C

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

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

Solution

$\begin{array}{l} \int {\{\frac{(\log x - 1)}{1 + (\log x)^{2}}\}}^{2} dx = \int \frac{(\log x)^{2} + 1 - 2 \log x}{{[(\log x)^{2} + 1]}^{2}} dx \\ = \int \frac{(\log x)^{2} + 1 - 2 x (\log x \cdot \frac{1}{x})}{{[(\log x)^{2} + 1]}^{2}} dx \\ = \int \frac{d}{dx} [\frac{x}{(\log x)^{2} + 1}] dx \\ = \frac{x}{(\log x)^{2} + 1} + C \end{array}$

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