JEE Mains · Maths · STD 12 - 7.2 definite integral
If the value of the integral \(\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} \,d x=\alpha e^{-1}+\beta\) where \(\alpha, \beta \in R, 5 \alpha+6 \beta=0\), and \([\mathrm{x}]\) denotes the greatest integer less than or equal to \(x\); then the value of \((\alpha+\beta)^{2}\) is equal to :
- A \(100\)
- B \(25\)
- C \(16\)
- D \(36\)
Answer & Solution
Correct Answer
(B) \(25\)
Step-by-step Solution
Detailed explanation
\(I=\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} \,d x\) \(\int_{0}^{1} \frac{x}{e^{x}} d x+\int_{1}^{2} \frac{x+1}{e^{x-1}} d x+\int_{2}^{3} \frac{x+2}{e^{x-2}} d x+\ldots . \int_{4}^{5} \frac{x+4}{e^{x-4}} \,d x\) \(\quad\quad\quad\quad\quad x=t+1\quad x=z+2 \quad\quad\quad x=y+4\)…
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