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

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

Answers (1)

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

Solution:

\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

dx=xdt

\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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