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Definite Integration question

2003 · Shift 0 · Q84
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Definite Integration question

2003 · Shift 0 · Q84

JEE MainMathematicsDefinite IntegrationMCQ+4 / −1
If f(a+b−x)=f(x)f\left( {a + b - x} \right) = f\left( x \right)f(a+b−x)=f(x) then ∫abxf(x)dx\int\limits_a^b {xf\left( x \right)dx}a∫b​xf(x)dx is equal to
  1. A
    a+b2∫abf(a+b+x)dx{{a + b} \over 2}\int\limits_a^b {f\left( {a + b + x} \right)dx}2a+b​a∫b​f(a+b+x)dx
  2. B
    a+b2∫abf(b−x)dx{{a + b} \over 2}\int\limits_a^b {f\left( {b - x} \right)dx}2a+b​a∫b​f(b−x)dx
  3. C
    a+b2∫abf(x)dx{{a + b} \over 2}\int\limits_a^b {f\left( x \right)dx}2a+b​a∫b​f(x)dx
  4. D
     b−a2∫abf(x)dx\,{{b - a} \over 2}\int\limits_a^b {f\left( x \right)dx}2b−a​a∫b​f(x)dx
View written solutionFree

Correct answer: C

  1. We are given the symmetry condition f(a+b−x)=f(x).f(a+b-x)=f(x).f(a+b−x)=f(x). We need to evaluate I=\int_a^b x f(x)\,dx.

  2. Use the standard substitution x=a+b−t⇒dx=−dt.x=a+b-t \quad \Rightarrow \quad dx=-dt.x=a+b−t⇒dx=−dt. When x=ax=ax=a, then t=bt=bt=b; when x=bx=bx=b, then t=at=at=a. So, I=∫abxf(x) dx=∫ba(a+b−t)f(a+b−t)(−dt).I=\int_a^b x f(x)\,dx=\int_b^a (a+b-t)f(a+b-t)(-dt).I=∫ab​xf(x)dx=∫ba​(a+b−t)f(a+b−t)(−dt). Reversing limits, I=∫ab(a+b−t)f(a+b−t) dt.I=\int_a^b (a+b-t)f(a+b-t)\,dt.I=∫ab​(a+b−t)f(a+b−t)dt. Using the given condition f(a+b−t)=f(t)f(a+b-t)=f(t)f(a+b−t)=f(t), I=∫ab(a+b−t)f(t) dt.I=\int_a^b (a+b-t)f(t)\,dt.I=∫ab​(a+b−t)f(t)dt. Renaming ttt back to xxx, I=∫ab(a+b−x)f(x) dx.I=\int_a^b (a+b-x)f(x)\,dx.I=∫ab​(a+b−x)f(x)dx.

  3. Now add the two expressions for III: I=∫abxf(x) dxI=\int_a^b x f(x)\,dxI=∫ab​xf(x)dx and I=∫ab(a+b−x)f(x) dx.I=\int_a^b (a+b-x)f(x)\,dx.I=∫ab​(a+b−x)f(x)dx. Therefore, 2I=\int_a^b \big[x+(a+b-x)\big]f(x)\,dx=\int_a^b (a+b)f(x)\,dx. So, 2I=(a+b)∫abf(x) dx.2I=(a+b)\int_a^b f(x)\,dx.2I=(a+b)∫ab​f(x)dx. Hence, I=a+b2∫abf(x) dx.I=\frac{a+b}{2}\int_a^b f(x)\,dx.I=2a+b​∫ab​f(x)dx.

  4. Compare with the options:

  • Option A: involves f(a+b+x)f(a+b+x)f(a+b+x), not obtained.
  • Option B: involves f(b−x)f(b-x)f(b−x), not generally correct.
  • Option C: a+b2∫abf(x) dx,\frac{a+b}{2}\int_a^b f(x)\,dx,2a+b​∫ab​f(x)dx, which matches exactly.
  • Option D: incorrect factor.

Therefore, the correct answer is Option C.

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