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Definite Integration question

2021 · 16 Mar · Shift 2 · Q30
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Definite Integration question

2021 · 16 Mar · Shift 2 · Q30

JEE MainMathematicsDefinite IntegrationMCQ+4 / −1
Consider the integral I=∫010[x]e[x]ex−1dxI = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx}I=∫010​ex−1[x]e[x]​dx, where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :
  1. A
    45 (e −-− 1)
  2. B
    45 (e + 1)
  3. C
    9 (e + 1)
  4. D
    9 (e −-− 1)
View written solutionFree

Correct answer: A

  1. Interpret the integrand carefully

The integral is

I=∫010[x]e[x]ex−1 dx,I=\int_0^{10} \frac{[x]e^{[x]}}{e^{x-1}}\,dx,I=∫010​ex−1[x]e[x]​dx,

where [x][x][x] is the greatest integer function.

So the integrand can be rewritten as

[x]e[x]ex−1=[x]e[x]−x+1.\frac{[x]e^{[x]}}{e^{x-1}}=[x]e^{[x]-x+1}.ex−1[x]e[x]​=[x]e[x]−x+1.
  1. Break the interval using the floor function

For x∈[n,n+1)x\in[n,n+1)x∈[n,n+1), where n=0,1,2,…,9n=0,1,2,\dots,9n=0,1,2,…,9, we have

[x]=n.[x]=n.[x]=n.

Hence on this interval,

[x]e[x]−x+1=nen−x+1.[x]e^{[x]-x+1}=ne^{n-x+1}.[x]e[x]−x+1=nen−x+1.

Therefore,

I=∑n=09∫nn+1nen−x+1 dx.I=\sum_{n=0}^{9}\int_n^{n+1} ne^{n-x+1}\,dx.I=n=0∑9​∫nn+1​nen−x+1dx.
  1. Evaluate each piece

For a fixed nnn,

∫nn+1nen−x+1 dx=nen+1∫nn+1e−x dx.\int_n^{n+1} ne^{n-x+1}\,dx = n e^{n+1}\int_n^{n+1} e^{-x}\,dx.∫nn+1​nen−x+1dx=nen+1∫nn+1​e−xdx.

Now,

∫nn+1e−x dx=[−e−x]nn+1=e−n−e−(n+1).\int_n^{n+1} e^{-x}\,dx = \left[-e^{-x}\right]_n^{n+1}=e^{-n}-e^{-(n+1)}.∫nn+1​e−xdx=[−e−x]nn+1​=e−n−e−(n+1).

So,

∫nn+1nen−x+1 dx=nen+1(e−n−e−(n+1)).\int_n^{n+1} ne^{n-x+1}\,dx = n e^{n+1}\left(e^{-n}-e^{-(n+1)}\right).∫nn+1​nen−x+1dx=nen+1(e−n−e−(n+1)).

Simplifying,

=n(e−1).= n(e-1).=n(e−1).

Thus,

I=∑n=09n(e−1)=(e−1)∑n=09n.I=\sum_{n=0}^{9} n(e-1)=(e-1)\sum_{n=0}^{9} n.I=n=0∑9​n(e−1)=(e−1)n=0∑9​n.
  1. Compute the sum
∑n=09n=9⋅102=45.\sum_{n=0}^{9} n = \frac{9\cdot 10}{2}=45.n=0∑9​n=29⋅10​=45.

Hence,

I=45(e−1).I=45(e-1).I=45(e−1).
  1. Match with the options
I=45(e−1)I=45(e-1)I=45(e−1)

which corresponds to Option A.

  1. Compare with stored correct answer

Stored correct answer: A

Our derived answer is also A, so they agree.

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