AP EAMCET · Maths · Definite Integration
\(\int_{\frac{2}{e}}^{\frac{1}{e}} \frac{1}{\left[x(\log x)^{\frac{1}{3}}\right]} d x\) is equal to
- A \(\frac{3}{2}\left\{1+(\log (2)-1)^{\frac{2}{3}}\right\}\)
- B \(1\)
- C \(\frac{3}{2}\left\{1+(\log (2)+1)^{\frac{3}{2}}\right\}\)
- D \(\frac{3}{2}\left\{1-(\log (2)-1)^{\frac{2}{3}}\right\}\)
Answer & Solution
Correct Answer
(D) \(\frac{3}{2}\left\{1-(\log (2)-1)^{\frac{2}{3}}\right\}\)
Step-by-step Solution
Detailed explanation
Let \(I-\int_{2 / e}^{1 / e} \frac{1}{x(\log x)^{1 / 3}} d x\) Let \(\log x=t\) \(\therefore\) Upper limit \(t=\log 1 / e=-1\) \(\Rightarrow \quad \frac{1}{x} d x=d t\) lower limit \(t=\log 2 / e\) \(\begin{aligned} & =\log 2-\log e \\ & =\log 2-1\end{aligned}\)…
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