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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 30

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Answer: \frac{\left ( \log x \right )^{1-n}}{1-n}+c

Hints: You must know about the integral rule of trigonometric functions

Given \int \frac{1}{x\left ( \log x \right )^{n}}dx

Solution:I=\int \frac{1}{x\left ( \log x \right )^{n}}dx

Let \log x=t   differentiate both sides,  \left [ \frac{d}{dx}\log x =\frac{1}{x}\right ]



\int \frac{1}{x\left ( \log x \right )^{n}}dx

Put\log x=t

\begin{aligned} &d x=x \cdot d t \\ &=\int \frac{1}{x \cdot t^{n}} \cdot x d t \\ &=\int \frac{1}{t^{n}} d t \\ &=\int t^{-n} d t \\ &=\frac{t^{-n+1}}{-n+1}+c \\ &=\frac{t^{1-n}}{1-n}+c \end{aligned}

=\frac{\left ( \log x \right )^{1-n}}{1-n}+c                                                               \left [ \int x^{n} dx = \frac{x^{n+1}}{n+1}\right ]

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