Integral Calculus
Integration & Limits
MMTS_Full_Test_02
Grade 12
Question:
$\int\dfrac{x(x\tan^{-1}x+(\ln x)(\ln(\ln x)))+\tan^{-1}x}{(x^3+x)\ln x}=f(x)+c$ where $f(e)=0$. Then $\left[\lim_{x\to 1^+}\dfrac{f(x)}{\tan\left(\frac{\pi x}{2}\right)}+\frac{11}{10}\right]=$
Step-by-Step Solution
Key Concept: Identify the integrand structure; use partial fractions
After integration and applying $f(e)=0$: $\lim_{x\to 1^+}\frac{f(x)}{\tan(\pi x/2)}+\frac{11}{10}=1$. GIF of limit near 1 gives answer 1.
Correct Answer: 2