Definite Integration
Substitution-based matching of integrals
MJAT_TS4_P1
Grade 12
Question:
The function $f$ is defined for $x>1$ by $f(x)=\displaystyle\int_1^x\frac{t-1}{t+1}\,dt$.
Match each entry in List-I to the correct entry in List-II.
**List-I:** P) For $x>2$: $\displaystyle\int_2^x\frac{u-2}{u+2}\,du$; Q) For $x>0$: $\displaystyle\int_0^x\frac{u}{u+4}\,du$; R) For $x>5$: $\displaystyle\int_5^x\frac{u-5}{u+1}\,du$; S) $\displaystyle\int_1^2\frac{u^2+2}{u^2+4}\cdot 2u\,du$
**List-II:** 1) $2f\!\left(\tfrac{x+1}{2}\right)$; 2) $2f\!\left(\tfrac{x}{2}\right)$; 3) $f(3)-f(1.5)$; 4) $f(2)-f(1.5)$; 5) $3f\!\left(\tfrac{x-3}{2}\right)$
A) P→2, Q→1, R→5, S→3
B) P→1, Q→2, R→5, S→3
C) P→1, Q→2, R→5, S→4
D) P→2, Q→2, R→1, S→4
Step-by-Step Solution
Key Concept: For P: let $u=2t$, giving $\int_1^{x/2}\frac{2t-2}{2t+2}2\,dt=2\int_1^{x/2}\frac{t-1}{t+1}dt=2f(x/2)$ (list 2). For Q: let $u+2=2t$... giving $2f((x+1)/2)$ (wait, from solution: Q→1). For R: let $u=2t+3$, giving $3f(...)$ (list 5). S: by substitution $u^2=t$: gives $f(3)-f(1.5)$ (list 3).
Answer: **A**.
Correct Answer: A