By property $\mathbf{P_3}$ of definite integrals, $\displaystyle\int_a^b f(x)\,dx$ equals
A$\displaystyle\int_a^b f(a-x)\,dx$
B$\displaystyle\int_a^b f(a+b-x)\,dx$
C$\displaystyle\int_0^b f(b-x)\,dx$
D$-\displaystyle\int_a^b f(a+b-x)\,dx$
Answer & Solution
Correct answer: B. $\displaystyle\int_a^b f(a+b-x)\,dx$
1. Property $\mathbf{P_3}$ states $\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx$.
2. It follows from substituting $t=a+b-x$ and applying $\mathbf{P_1}$ and $\mathbf{P_0}$.
3. Hence option B is correct; option A omits the $b$ term.
_Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.342_
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