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  • please solve rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 3 maths textbook solution

please solve rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 3 maths textbook solution

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\frac{2}{7} a x^{\frac{7}{2}}+\frac{2}{5} b x^{\frac{5}{2}}+\frac{2}{3} c x^{\frac{3}{2}}+c

To solve this we multiply \sqrt{x} by ax^{2}+bx+c

\begin{aligned} &\text { Given: } \int\left\{\sqrt{x}\left(a x^{2}+b x+c\right)\right\} d x \\ &\text { Solution: } \int\left\{\sqrt{x}\left(a x^{2}+b x+c\right)\right\} d x \end{aligned}

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

\begin{aligned} &=a \frac{x^{\frac{5}{2}+1}}{\frac{5}{2}+1}+b \frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+c \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1} \quad\left\{\int x^{n} d x=\frac{x^{n+1}}{n+1}+c\right\} \\ &=\frac{2}{7} a x^{\frac{7}{2}}+\frac{2}{5} b x^{\frac{5}{2}}+\frac{2}{3} c x^{\frac{3}{2}}+c \end{aligned}

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