Definite Integration
King's Property / Symmetry
MJMT_Full_Test_05
Grade 12
Question:
If $\displaystyle\sum_{i=1}^{n}(\sin^{-1} x_i + \cos^{-1} y_i) = \frac{9\pi}{...}$, evaluate $\displaystyle\int_{-1}^{1} x\ln(1+x^2)\cdot\frac{e^x}{1+e^{2x}}\,dx$
Step-by-Step Solution
Key Concept: The integrand is an odd function; integral of odd function over symmetric interval is 0.
Let $g(x)=x\ln(1+x^2)\cdot\frac{e^x}{1+e^{2x}}$. $g(-x) = -x\ln(1+x^2)\cdot\frac{e^{-x}}{1+e^{-2x}} = -x\ln(1+x^2)\cdot\frac{1}{e^x+e^{-x}}\cdot\frac{e^x}{1} = -g(x)$ after checking. So $g$ is odd and the integral $= 0$.
Correct Answer: 1