Definite Integration
Definite Integral with Inverse Hyperbolic Substitution
nta_pyq_2023_apr
Grade None
Question:
The value of the integral $\displaystyle\int_{-\log_e 2}^{\log_e 2} e^x\!\left(\log_e(e^x+\sqrt{1+e^{2x}})\right)dx$ is equal to
$\log_e\!\left(\dfrac{\sqrt{2}(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)-\dfrac{\sqrt{5}}{2}$
$\log_e\!\left(\dfrac{(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)+\dfrac{\sqrt{5}}{2}$
$\log_e\!\left(\dfrac{2(2+\sqrt{5})}{\sqrt{1+\sqrt{5}}}\right)-\dfrac{\sqrt{5}}{2}$
$\log_e\!\left(\dfrac{\sqrt{2}(3-\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}\right)+\dfrac{\sqrt{5}}{2}$
Step-by-Step Solution
Key Concept: Substitute $e^x=t$, limits $\frac{1}{2}$ to $2$. Integrate by parts with $u=\ln(t+\sqrt{1+t^2})$, $dv=dt$.
$I=\ln\frac{\sqrt{2}(2+\sqrt{5})^2}{\sqrt{1+\sqrt{5}}}-\frac{\sqrt{5}}{2}$.
Correct Answer: 1