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Evaluate \(\rm \int_0^4 \frac{1}{1+ \sqrt x}dx\)
1. 2 - 2log 3
2. 4 - log 3
3. 4 - 2log 3
4. None of the above
5. 4 + 2log 3

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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

x 0 4
t 1 3

 

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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