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Integration (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Integration (p-1)2 Marks
Question
Evalute : $\int\left[\frac{1}{\log x}-\frac{1}{(\log x)^2}\right] d x$
✓

Answer

Let $I=\int\left[\frac{1}{\log x}-\frac{1}{(\log x)^2}\right] d x$
Put $\log x=t \quad \therefore x=e^t$
$\therefore d x=e^t d t$
$
\therefore I=\int\left(\frac{1}{t}-\frac{1}{t^2}\right) e^t d t
$
Let $f(t)=\frac{1}{t}$. Then $f^{\prime}(t)=-\frac{1}{t^2}$
$
\begin{aligned}
\therefore I & =\int e^t\left[f(t)+f^{\prime}(t)\right] d t \\
& =e^t \cdot f(t)+c=e^t \times \frac{1}{t}+c \\
& =\frac{x}{\log x}+c
\end{aligned}
$

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