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  • Provide solution for rd sharma maths class 12 chapter Indefinite Integrals exercise 18.2 question 19

Provide solution for rd sharma maths class 12 chapter Indefinite Integrals exercise 18.2 question 19

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\frac{3x^{2}}{2}+\frac{2x^{3}}{3}+c

Hint: To solve this equation we do middle term equation

\begin{aligned} &\text { Given: } \int \frac{2 x^{4}+7 x^{3}+6 x^{2}}{x^{2}+2 x} d x \\ &\text { Solution: } \int \frac{2 x^{4}+4 x^{3}+3 x^{3}+6 x^{2}}{x(x+2)} d x \\ &=\int \frac{2 x^{3}(x+2)+3 x^{2}(x+2)}{x(x+2)} d x \\ &=\int \frac{2 x^{3}+3 x^{2}}{x} d x \\ &=\int \frac{2 x^{3}+3 x^{2}}{x} d x \\ &=\int \frac{x\left(2 x^{2}+3 x\right)}{x} d x \\ &=\int 3 x+2 x^{2} d x\left[\int x^{a} d x=\frac{1}{a} x^{a+1}+c, a \neq-1\right] \\ &=3 \frac{x^{1+1}}{1+1}+2 \frac{x^{2+1}}{2+1}+c \\ &=\frac{3 x^{2}}{2}+\frac{2 x^{3}}{3}+c \end{aligned}

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