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Find area bounded by the curve `y = lnx + tan^(-1)x` and x-axis between ordiantes `x = 1` and `x = 2`. |
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Answer» `y = ln x + tan^(-1)x` Domain `x gt 0 , (dy)/(dx) = 1/x + (1)/(1+x^(2)) gt 0` `y` is increasing and `x = 1, y = (pi)/(4) rArr y` is positive in `[1,2]` y in increasing and `x = 1, y = (pi)/(4) rArr y` is positive in `[1,2]` `:.` Required area `= underset(1)overset(2)int(lnx+tan^(-1)) dx = [xlnx - x + x tan^(-1)- 1/2ln(1+x^(2))]_(1)^(2)` `= 2ln2-2+2tan^(-1)2-1/2 ln5-0+1-tan^(-1)1/2 ln 2` `= 5/2 ln 2- 1/2 ln 5+2 tan^(-1) - (pi)/(4) - 1` |
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