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Evaluate \(\rm \int_0^4 \frac{1}{1+ \sqrt x}dx\)1. 2 - 2log 32. 4 - log 33. 4 - 2log 34. None of the above |
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Answer» Correct Answer - Option 3 : 4 - 2log 3 Concept: \(\rm \int \frac{1}{x} dx = \log x + c\) Calculation: I = \(\rm \int_0^4 \frac{1}{1+ \sqrt x}dx\) Let 1 + \(\rm \sqrt x\) = t .... (1) Differentiating with respect to x, we get \(\rm \Rightarrow (0+\frac{1}{2\sqrt x})dx = dt\) \(\rm \Rightarrow dx = {2\sqrt x}dt\) From equation (1), we get \(\rm \sqrt x\) = t - 1 ∴ dx = 2(t - 1)dt
Now, I = \(\rm \int_1^3 \frac{2(t-1)}{t}dt \) = \(\rm 2\int_1^3 \left(1-\frac{1}{t} \right )dt \) = \(\rm 2\left[t - \log t \right ]_1^3\) = 2 [(3 - log 3) - (1 - log 1)] = 2(2 - log 3) = 4 - 2log 3 |
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