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

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

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\frac{x^{5}}{5}-\frac{x^{3}}{3}+x+c

Hint: To solve this equation we usea^{3}+b^{3} formula , then integral it.

\begin{aligned} &\text { Given: } \int \frac{x^{6}+1}{x^{2}+1}\\ &\text { Solution: } x^{6}+1\\ &=\left(x^{2}\right)^{3}+\left(1^{2}\right)^{3} \quad\left[a^{3}+b^{3}=(a+b)\left(a^{2}-a b-b^{2}\right)\right]\\ &=\left(x^{2}+1\right)\left(x^{4}-x^{2}+1\right)\\ &\Rightarrow \int \frac{\left(x^{2}+1\right)\left(x^{4}-x^{2}+1\right)}{x^{2}+1} d x\\ &=\int\left(x^{4}-x^{2}+1\right) d x\\ &\left[\because \int x^{a}=\frac{1}{a} x^{a+1}+c, a \neq-1\right]\\ &\Rightarrow \frac{x^{4+1}}{4+1}-\frac{x^{2+1}}{2+1}+\frac{x^{0+1}}{0+1}+c\\ &\Rightarrow \frac{x^{5}}{5}-\frac{x^{3}}{3}+x+c \end{aligned}

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