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Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.3  Question 7 Maths Textbook Solution.

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Answer: \frac{1}{2} \log |2 x+1|+\frac{1}{2(2 x+1)}+c

Hint : \text { To solve this equation we differentiate differently }

Given: \int \frac{2 x}{(2 x+1)^{2}} d x

Solution: \int \frac{2 x}{(2 x+1)^{2}} d x

\int \frac{(2 x+1)-1}{(2 x+1)^{2}} d x

=\int \frac{1}{(2 x+1)} d x-\int \frac{1}{(2 x+1)^{2}} d x

\left[\begin{array}{l} \int \frac{1}{a x+b} d x=\log \frac{1}{a}|a x+b|+c \\ \int(a x+b)^{n}=\frac{(a x+b)^{n+1}}{a(n+1)}+c, n \neq 1 \end{array}\right]

\begin{aligned} &=\frac{1}{2} \log |2 x+1|+\frac{1}{2} \frac{1}{(2 x+1)}+c \\ &=\frac{1}{2} \log |2 x+1|+\frac{1}{2(2 x+1)}+c \end{aligned}


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