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  • Provide solution for rd sharma maths class 12 chapter Indefinite Integrals exercise 18.2 question 16

Provide solution for rd sharma maths class 12 chapter Indefinite Integrals exercise 18.2 question 16

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\frac{2}{5} x^{\frac{5}{2}}-\frac{2}{3} x^{\frac{3}{2}}-4 \sqrt{x}+c

Hint: To solve this equation we multiply the upper term

\begin{aligned} &\text { Given: } \int \frac{(x+1)(x-2)}{\sqrt{x}} d x \\ &\text { Solution: } \int \frac{x(x-2)+1(x+2)}{\sqrt{x}} d x \\ &=\int \frac{x^{2}-2 x+x+2}{\sqrt{x}} d x \\ &=\int \frac{x^{2}-x+2}{\sqrt{x}} d x \\ &=\int \frac{x^{2}}{\sqrt{x}}-\frac{x}{\sqrt{x}}-\frac{2}{\sqrt{x}} d x \end{aligned}

\begin{aligned} &=\int x^{\frac{3}{2}} d x-\int \sqrt{x} d x-2 \int d x \\ &{\left[\because \int x^{n} d x=\frac{x^{n+1}}{n+1}+c, n \neq-1\right]} \\ &=2 \frac{x^{\frac{5}{2}}}{5}-2 \frac{x^{\frac{3}{2}}}{3}-4 \sqrt{x}+c \\ &=\frac{2}{5} x^{\frac{5}{2}}-\frac{2}{3} x^{\frac{3}{2}}-4 \sqrt{x}+c \end{aligned}

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