MHT CET · Maths · Indefinite Integration
\(\int \frac{5\left(x^6+1\right)}{x^2+1} \mathrm{~d} x=\) (Where \(C\) is a constant of integration.)
- A \(5\left(x^7+1\right)+\log \left(x^2+1\right)+C\)
- B \(x^5-\frac{5 x^3}{3}+5 x+C\)
- C \(\frac{5 x^7}{7}+5 x+5 \tan ^{-1} x+C\)
- D \(5 \tan ^{-1} x+\log \left(x^2+1\right)+C\)
Answer & Solution
Correct Answer
(B) \(x^5-\frac{5 x^3}{3}+5 x+C\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \int \frac{5\left(x^6+1\right)}{x^2+1} \mathrm{~d} x=\int \frac{5\left(x^2+1\right)\left(x^4-x^2+1\right)}{x^2+1} \mathrm{~d} x \\ & =5 \int\left(x^4-x^2+1\right) \mathrm{d} x \\ & =5\left\{\frac{x^5}{5}-\frac{x^3}{3}+x\right\}+C \\ & =x^5-\frac{5}{3} x^3+5 x+C\end{aligned}\)
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