Definite Integration
Multiple-Correct — Properties
Grade 12
Question:
<p>If \(g(x)=f(a+b-x)\) and \(I=\int_a^b f(x)\,dx\), \(J=\int_a^b g(x)\,dx\), which are correct? [JEE Advanced 2013]</p>
I=J
I-J=0
I+J=f(a+b) \cdot (b-a)
I\neqJ in general
Step-by-Step Solution
Key Concept: Sub x \to a+b-x in J: J = \intₐ^b f(a+b-(a+b-x))dx = \intₐ^b f(x)dx = I.
<div class='solution'>
<p>In $J=\int_a^b g(x)dx=\int_a^b f(a+b-x)dx$, substitute $u=a+b-x$, $du=-dx$:</p>
<p>$$J=\int_b^a f(u)(-du)=\int_a^b f(u)du=I$$</p>
<p>So $I=J$ always. ✓(A, B)</p>
</div>
Correct Answer: A