First slide

Introduction to integration

Question

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

Moderate

A

$x \log (\log x) + x (\log x)^{- 1} + c$
B

$x \log (\log x) - (\log x)^{- 1} + c$
C

$x \log (\log x) - x (\log x)^{- 1} + c$
D

None of these

Solution

$\begin{array}{l} \int (\log (\log x) + \frac{1}{(\log x)^{2}}) d x \\ = \int 1 \cdot \log (\log x) d x + \int \frac{d x}{(\log x)^{2}} (a p p l y u v r u l e t o t h e f i r s t i n t e g r a n d) \\ = x \log (\log x) - \int \frac{1}{x \log x} x d x + \int \frac{d x}{(\log x)^{2}} + c \\ = x \log (\log x) - \int (\log x)^{- 1} d x + \int \frac{d x}{(\log x)^{2}} + c \\ = x \log (\log x) - [x (\log x)^{- 1} + \int (\log x)^{- 2} d x] + \int (\log x)^{- 2} d x \\ = x \log (\log x) - x (\log x)^{- 1} + c \end{array}$

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