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Please Solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise18.5 Question 2 Maths Textbook Solution.

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Answer:  \frac{2}{5}(x+2)^{\frac{5}{2}}-\frac{6}{3}(x+2)^{\frac{1}{2}}+C

Hint: Let u= x+2

Given: \int x\sqrt{x+2}dx


Let u= x+2

\frac{du}{dx}= 1

Applying the substitution we have:

\begin{aligned} &I=\int(u-2) \sqrt{u} d u \\ &\therefore I=(u-2) u^{\frac{1}{2}} d u \\ &=\int u^{\frac{3}{2}}-2 u^{\frac{1}{2}} d u \end{aligned}
\begin{aligned} &\therefore I=\frac{u^{\frac{5}{2}}}{\frac{5}{2}}-\frac{2 u^{\frac{3}{2}}}{\frac{3}{2}}+C \\ &\therefore I=\frac{2}{5} u^{\frac{5}{2}}-\frac{4}{3} u^{\frac{3}{2}}+C \\ &\therefore I=\frac{2}{5}(x+2)^{\frac{5}{2}}-\frac{4}{3}(x+2)^{\frac{3}{2}}+C \end{aligned}


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