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  • Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.10 Question 10

Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.10 Question 10

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Answer:  3 x^{\frac{1}{3}}+3 \log \left|x^{\frac{1}{3}}-1\right|+c

Hint: Use substitution method to solve this type of integral

Given:  \int \frac{1}{x^{\frac{1}{3}}\left(x^{\frac{1}{3}}-1\right)} d x

Solution: let  I=\int \frac{1}{x^{\frac{1}{3}}\left(x^{\frac{1}{3}}-1\right)} d x

Substitute  x=t^{3} \Rightarrow d x=3 t^{2} d t   then

\begin{aligned} I &=I=\int \frac{1}{t(t-1)} 3 t^{2} d t \quad\left(\because x^{\frac{1}{3}}=t\right) \\ & \end{aligned}

=3 \int \frac{t^{2}}{t(t-1)} d t=3 \int \frac{t}{t-1} d t \\

=3 \int \frac{t+1-1}{t-1} d t=3 \int\left(\frac{t-1}{t-1}+\frac{1}{t-1}\right)dt

\begin{aligned} &=3 \frac{t^{0+1}}{0+1}+3 \log |t-1|+c \qquad\left[\int \frac{1}{t} d t=\log |t|+c \& \int t^{n} d t=\frac{t^{n+1}}{n+1}+c\right] \\ & \end{aligned}

=3 t+3 \log |t-1|+c \\

\therefore I=3 x^{\frac{1}{3}}+3 \log \left|x^{\frac{1}{3}}-1\right|+c

 

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