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  • Provide solution for rd sharma maths class 12 chapter Indefinite Integrals exercise 18.2 question 20

Provide solution for rd sharma maths class 12 chapter Indefinite Integrals exercise 18.2 question 20

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\begin{aligned} & \frac{5 x^{3}}{3}+\frac{7 x^{2}}{2}+c\\ &\text { Hint: To solve this equation we do middle term spilt }\\ &\text { Given: } \int \frac{5 x^{4}+12 x^{3}+7 x^{2}}{x^{2}+x} d x \end{aligned}\begin{aligned} &\text { Solution: } \int \frac{5 x^{4}+12 x^{3}+7 x^{2}}{x^{2}+x} d x \\ &=\int \frac{x^{2}\left(5 x^{2}+12 x+7\right)}{x(x+1)} d x \\ &=\int \frac{x\left(5 x^{2}+12 x+7\right)}{(x+1)} d x \\ &=\int \frac{x\left(5 x^{2}+5 x+7 x+7\right)}{(x+1)} d x \\ &=\int \frac{x(5 x(x+1)+7(x+1))}{(x+1)} d x \\ &=\int \frac{x(5 x+7)(x+1)}{(x+1)} d x \\ &=\int x(5 x+7) d x \\ &=\int 5 x^{2}+7 x d x\left[\int x^{a} d x=\frac{1}{a} x^{a+1}+c, a \neq-1\right] \\ &=5 \frac{x^{2+1}}{2+1}+7 \frac{x^{1+1}}{1+1}+c \\ &=\frac{5 x^{3}}{3}+\frac{7 x^{2}}{2}+c \end{aligned}

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