≤ft( 1/ x - 1/( x)² ) \, dx [MTP-APRIL -19]

Correct Answer is : xlog⁡x+cx/ x + clogxx+c

Correct Answer is : xlog⁡x+cx/ x + clogxx+c

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Integral Calculus

Question : 64 of 73

Marks: +1, -0

\int \left( \frac{1}{\log x} - \frac{1}{(\log x)^2} \right) \, dx

[MTP-APRIL -19]

Solution:

Concept:

The derivative of

\frac{x}{\log x}

gives the integrand.

Explanation:

Let

f(x) = \frac{x}{\log x}

Compute

f'(x)

using quotient rule:

f'(x) = \frac{\log x \cdot 1 - x \cdot \frac{1}{x}}{(\log x)^2} = \frac{\log x - 1}{(\log x)^2} = \frac{1}{\log x} - \frac{1}{(\log x)^2}

Thus, the integrand is exactly the derivative of

\frac{x}{\log x}

Therefore, the integral is

\frac{x}{\log x} + c

Answer:

\frac{x}{\log x} + c

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