∫dxx(log⁡x)m is equal to - Infinity Learn by Sri Chaitanya

A
$\frac{(\log x)^{m}}{m} + C$
B
$\frac{(\log x)^{m - 1}}{m - 1} + C$
C
$\frac{(\log x)^{1 - m}}{1 - m} + C$
D
$\frac{(\log x)^{1 - m}}{m} + C$

Solution:

Here, given integrand is not in any standard form. So,we convert it by substitution method in a standard form and then integrate it.

$\begin{array}{l} \int \frac{1}{x (\log x)^{m}} dx \\ Let \log x = t \Rightarrow \frac{1}{x} = \frac{dt}{dx} \Rightarrow dx = xdt \\ ∴ \begin{aligned} \int \frac{1}{x (\log x)^{m}} dx = & \int \frac{1}{x (t)^{m}} xdt = \int t^{- m} dt \\ = & \frac{t^{- m + 1}}{- m + 1} + C = \frac{(\log x)^{1 - m}}{1 - m} + C \end{aligned} \end{array}$

$\int \frac{dx}{x (\log x)^{m}}$ is equal to

Solution:

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