Definite Integration
Symmetric Integral with Exponential Denominator
nta_pyq_2023_apr
Grade 12
Question:
If $\displaystyle\int_0^1\dfrac{1}{(5+2x-2x^2)(1+e^{2-4x})}\,dx=\dfrac{1}{\alpha}\log_e\!\left(\alpha+\dfrac{1}{\beta}\right)$, $\alpha,\beta>0$, then $\alpha^4-\beta^4$ is equal to
Step-by-Step Solution
Key Concept: Substitute $t=x-\frac{1}{2}$ to symmetrise the exponential. Use the property $\frac{1}{1+e^{-4t}}+\frac{1}{1+e^{4t}}=1$ to simplify to $\frac{1}{2}\int_{-1/2}^{1/2}\frac{dt}{\frac{11}{4}-t^2}$.
$\alpha=\sqrt{11},\ \beta=\sqrt{10}$. $\alpha^4-\beta^4=21$.
Correct Answer: 4