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#### please solve rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 7 maths textbook solution

$2 \sqrt{x}+\frac{2}{7} x^{\frac{7}{2}}+2 x^{\frac{3}{2}}+\frac{6}{5} x^{\frac{5}{2}}+c$

Hint: To solve this equation we use $(a+b)^{3}$ formula then find the integral

\begin{aligned} &\text { Given: } \int \frac{(1+x)^{3}}{\sqrt{x}} d x \\ &\text { Solution: } I=\int \frac{(1+x)^{3}}{\sqrt{x}} d x \end{aligned}

\begin{aligned} &\text { Using identity }\\ &\left\{(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)\right\}\\ &=\int \frac{1+x^{3}+3 x(1+x)}{\sqrt{x}} d x\\ &=\int \frac{1+x^{3}+3 x+3 x^{2}}{\sqrt{x}} d x\\ &=\int x^{\frac{-1}{2}}+x^{\frac{5}{2}}+3 x^{\frac{1}{2}}+3 x^{\frac{3}{2}} d x \end{aligned}

\begin{aligned} &\text { Using identity }\\ &\left\{\int x^{a} d x=\frac{1}{a+1} x^{a+1}+c, a \neq-1\right\}\\ &=\frac{x^{\frac{-1}{2}}}{\frac{-1}{2}+1}+\frac{x^{\frac{5}{2}}+1}{\frac{5}{2}+1}+\frac{3 x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\frac{3 x^{\frac{3}{2}+1}}{\frac{3}{2}+1}\\ &=\frac{\sqrt{x}}{\frac{1}{2}}+\frac{x^{\frac{7}{2}}}{\frac{7}{2}}+\frac{3 x^{\frac{3}{2}}}{\frac{3}{2}}+\frac{3 x^{\frac{5}{2}}}{\frac{5}{2}}\\ &=2 \sqrt{x}+\frac{2}{7} x^{\frac{7}{2}}+\frac{6}{3} x^{\frac{3}{2}}+\frac{6}{5} x^{\frac{5}{2}}+c\\ &=2 \sqrt{x}+\frac{2}{7} x^{\frac{7}{2}}+2 x^{\frac{3}{2}}+\frac{6}{5} x^{\frac{5}{2}}+c \end{aligned}