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Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise Very Short Answers Question 39

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Answer: \frac{1}{3}\log \left | x^{3}+6x^{2}+5 \right |+c

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

Given: \int \frac{x^{2}+4x}{x^{3}+6x^{2}+5}dx


\int \frac{x^{2}+4x}{x^{3}+6x^{2}+5}dx

Put x^{3}+6x^{2}+5=t   and differentiate both sides,  \left [ \frac{d}{dx} x^{n}=nx^{n-1}\right ]

\begin{aligned} &3 x^{2}+12 x d x=d t \\ &3\left(x^{2}+4 x\right) d x=d t \\ &\left(x^{2}+4 x\right) d x=\frac{d t}{3} \\ &\int \frac{d t}{3 t}=\frac{1}{3} \int \frac{d t}{t} \\ &=\frac{1}{3} \log |t|+c \\ &=\frac{1}{3} \log \left|x^{3}+6 x^{2}+5\right|+c \quad\quad\quad\quad\quad\quad\quad &{\left[\int \frac{1}{x} d x=\log x+c\right]} \end{aligned}

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