Indefinite Integration
Log Substitution — IBP with cot
nta_pyq_2026_jan
Grade 12
Question:
Let $f(t)=\displaystyle\int\left(\frac{1-\sin(\log_e t)}{1-\cos(\log_e t)}\right)dt$, $t>1$. If $f(e^{\pi/2})=-e^{\pi/2}$ and $f(e^{\pi/4})=\alpha e^{\pi/4}$, then $\alpha$ equals
$1+\sqrt{2}$
$-1-\sqrt{2}$
$-1-2\sqrt{2}$
$-1+\sqrt{2}$
Step-by-Step Solution
Key Concept: Let $u=\log t$, $dt=e^u du$. $f=\int\tfrac{1-\sin u}{1-\cos u}e^u du$. Express $\tfrac{1-\sin u}{1-\cos u}=\tfrac{1}{2}\csc^2(u/2)-\cot(u/2)$. Recognise $e^u[g(u)+g'(u)]$ form with $g(u)=-\cot(u/2)$... Using IBP: $f(t)=t\cot\!\left(\tfrac{\log t}{2}\right)+C$.
$\alpha=-1-\sqrt{2}$.
Correct Answer: 2