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 $$\frac{5\ln(2)}{2} - \frac{\ln(5)}{2} + 2 \tan^{-1} 2 - \frac{\pi}{4} - 1 $$

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    $\begingroup$ What exactly is the bounded area defined by? $\endgroup$ – Toby Mak May 11 at 12:29
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    $\begingroup$ @TobyMak I'm really sorry, I forgot to mention, I have edited the uestion $\endgroup$ – Hema May 11 at 15:09

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.

  • $\begingroup$ I had actually forgotten to mention,sorry, I have edited the question now $\endgroup$ – Hema May 11 at 15:10
  • $\begingroup$ @Hema OK; edited. $\endgroup$ – J.G. May 11 at 15:48

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