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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 43

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Answer: \frac{1}{3}{\left ( 1+\log x \right )^{3}}+c

Hint: You must know about the integral rule of logarithmic functions.

Given: \int \frac{\left ( 1+\log x \right )^{2}}{x}dx


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

Let1+\log x = t     and differentiate both sides,  \frac{d}{dx}\log x=\frac{1}{x}

\begin{aligned} &0+\frac{1}{x} d x=d t \\ &\frac{1}{x}=\frac{d t}{d x} \\ &d x=x d t \end{aligned}

\therefore \int \frac{\left ( 1+\log x \right )^{2}}{x}dx

Put 1+\log x = t


\begin{aligned} &\int \frac{t^{2} x}{x} d t \\ &\int t^{2} d t \\ &=\frac{t^{2+1}}{2+1}+c \\ &=\frac{t^{3}}{3}+c \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\quad\left[\int x^{n} d x=\frac{x^{n+1}}{n+1}\right] \\ &=\frac{1}{3}(1+\log x)^{3}+c \end{aligned}

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