I'm not really sure how to do this without having a real graph using a graphing calculator. Adding the integrals would no work since some areas are overlapping, and even subtracting doesn't since one area is not within the other. Is there any quick way in which I can solve this in 3-4 minutes? The question is to find area bounded by given curve and x axis from x = 1 to x = 2.
The answer is $$\frac{5\ln(2)}{2} - \frac{\ln(5)}{2} + 2 \tan^{-1} 2 - \frac{\pi}{4} - 1 $$
Integrating by parts, $\int_1^2y(x)dx=[x\ln x-x+x\arctan x-\frac12\ln(1+x^2)]_1^2$ gives the desired value, although the given order of terms may not make this obvious.