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Please Solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise18.5 Question 9 Maths Textbook Solution.

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Answer: \frac{1}{3}(2 x-1)^{\frac{3}{2}}(3 x+4)+C

Hint: Let  5x+3=\lambda \left ( 2x-1\right )+\mu

Given: \int \left ( 5x+3 \right )\sqrt{2x-1}dx

Solution: Comparing the coefficient, we get
\begin{aligned} &\Rightarrow 2 \lambda=5 \Rightarrow \lambda=\frac{5}{2} \Rightarrow-\lambda+\mu=3 \\ &\Rightarrow \frac{-5}{2}+\mu=3 \Rightarrow \mu=3+\frac{5}{2} \Rightarrow \mu=\frac{11}{2} \\ &=\int[\lambda(2 x-1)+\mu] \sqrt{2 x-1} d x \Rightarrow \int\left[\frac{5}{2}(2 x-1)+\frac{11}{2}\right] \sqrt{2 x-1} d x \end{aligned}
\begin{aligned} &\therefore I=\int(5 x+3) \sqrt{2 x-1} d x \Rightarrow \frac{5}{2} \int(2 x-1)^{1+\frac{1}{2}} d x+\frac{11}{2} \int(2 x-1)^{\frac{1}{2}} d x \\ &\Rightarrow \frac{5}{2} \int(2 x-1)^{\frac{3}{2}}+\frac{11}{2} \int(2 x-1)^{\frac{1}{2}} d x \\ &\Rightarrow \frac{5}{2} \frac{(2 x-1)^{\frac{3}{2}+1}}{2\left[\frac{3}{2}+1\right]}+\frac{11}{2} \frac{(2 x-1)^{\frac{1}{2}+1}}{2\left[\frac{1}{2}+1\right]}+C \end{aligned}
\begin{aligned} &\Rightarrow \frac{5}{2} \frac{(2 x-1)^{\frac{5}{2}}}{2\left[\frac{5}{2}\right]}+\frac{11}{2} \frac{(2 x-1)^{\frac{3}{2}}}{2\left[\frac{3}{2}\right]}+C \Rightarrow \frac{5}{2} \frac{(2 x-1)^{\frac{5}{2}}}{5}+\frac{11}{2} \frac{(2 x-1)^{\frac{3}{2}}}{3}+C \\ &\Rightarrow \frac{1}{2}(2 x-1)^{\frac{5}{2}}+\frac{11}{2}(2 x-1)^{\frac{3}{2}}+C \Rightarrow \frac{1}{6}(2 x-1)^{\frac{3}{2}}[3(2 x-1)+11]+C \end{aligned}
\begin{aligned} &\Rightarrow \frac{1}{6}(2 x-1)^{\frac{3}{2}}[6 x-3+11]+C \Rightarrow \frac{1}{6}(2 x-1)^{\frac{3}{2}}[6 x+8]+C \\ &\Rightarrow \frac{1}{6}(2 x-1)^{\frac{3}{2}}[2(3 x+4)]+C \Rightarrow \frac{2}{6}(2 x-1)^{\frac{3}{2}}[3 x+4]+C \\ &\Rightarrow \frac{1}{3}(2 x-1)^{\frac{3}{2}}(3 x+4)+C \end{aligned}

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