Definite Integration
Tan Power Integral Using Integration by Parts
nta_pyq_2023_apr
Grade 12
Question:
The value of $\dfrac{e^{-\pi/4}+\displaystyle\int_0^{\pi/4}e^{-x}\tan^{50}x\,dx}{\displaystyle\int_0^{\pi/4}e^{-x}(\tan^{49}x+\tan^{51}x)\,dx}$
Step-by-Step Solution
Key Concept: Integrate $\int_0^{\pi/4}e^{-x}\tan^{50}x\,dx$ by parts: differentiate $\tan^{50}x$ and integrate $e^{-x}$. Use $\sec^2x=1+\tan^2x$.
$\frac{50I_1}{I_1}=50$.
Correct Answer: 2