The value of the integral - 1 2 log e ( x + x 2 + 1 ) d x [2024]

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Q.
The value of the integral $\int_{- 1}^{2} \log_{e} (x + \sqrt{x^{2} + 1}) d x$ [2024]

1 $\sqrt{5} - \sqrt{2} + \log_{e} (\frac{9 + 4 \sqrt{5}}{1 + \sqrt{2}})$

2 $\sqrt{2} - \sqrt{5} + \log_{e} (\frac{7 + 4 \sqrt{5}}{1 + \sqrt{2}})$

3 $\sqrt{2} - \sqrt{5} + \log_{e} (\frac{9 + 4 \sqrt{5}}{1 + \sqrt{2}})$

4 $\sqrt{5} - \sqrt{2} + \log_{e} (\frac{7 + 4 \sqrt{5}}{1 + \sqrt{2}})$

Ans.
(3)

Let $I = \int_{- 1}^{2} \log_{e} (x + \sqrt{x^{2} + 1}) d x$

$= {[\log_{e} (x + \sqrt{x^{2} + 1}) x]}_{- 1}^{2} - \int_{- 1}^{2} \frac{1}{(x + \sqrt{x^{2} + 1})} (1 + \frac{x}{\sqrt{x^{2} + 1}}) \cdot x d x$

$= 2 \log_{e} (2 + \sqrt{5}) + \log_{e} (\sqrt{2} - 1) - \int_{- 1}^{2} \frac{x}{\sqrt{x^{2} + 1}} d x$

$= \log [{(2 + \sqrt{5})}^{2} (\sqrt{2} - 1)] - {[\sqrt{x^{2} + 1}]}_{- 1}^{2}$

$= \log [\frac{9 + 4 \sqrt{5}}{1 + \sqrt{2}}] - \sqrt{5} + \sqrt{2}$

Solution

Q.
The value of the integral $\int_{- 1}^{2} \log_{e} (x + \sqrt{x^{2} + 1}) d x$ [2024]

1 $\sqrt{5} - \sqrt{2} + \log_{e} (\frac{9 + 4 \sqrt{5}}{1 + \sqrt{2}})$

2 $\sqrt{2} - \sqrt{5} + \log_{e} (\frac{7 + 4 \sqrt{5}}{1 + \sqrt{2}})$

3 $\sqrt{2} - \sqrt{5} + \log_{e} (\frac{9 + 4 \sqrt{5}}{1 + \sqrt{2}})$

4 $\sqrt{5} - \sqrt{2} + \log_{e} (\frac{7 + 4 \sqrt{5}}{1 + \sqrt{2}})$

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Solution

Q. The value of the integral ∫-12loge(x+x2+1)dx [2024]

1 5-2+loge(9+451+2)

2 2-5+loge(7+451+2)

3 2-5+loge(9+451+2)

4 5-2+loge(7+451+2)

Ans. (3) Let I=∫-12loge(x+x2+1)dx =[loge(x+x2+1)x]-12-∫-121(x+x2+1)(1+xx2+1)·x dx =2loge(2+5)+loge(2-1)-∫-12xx2+1dx =log[(2+5)2(2-1)]-[x2+1]-12 =log[9+451+2]-5+2

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Q.
The value of the integral $\int_{- 1}^{2} \log_{e} (x + \sqrt{x^{2} + 1}) d x$ [2024]

1 $\sqrt{5} - \sqrt{2} + \log_{e} (\frac{9 + 4 \sqrt{5}}{1 + \sqrt{2}})$

2 $\sqrt{2} - \sqrt{5} + \log_{e} (\frac{7 + 4 \sqrt{5}}{1 + \sqrt{2}})$

3 $\sqrt{2} - \sqrt{5} + \log_{e} (\frac{9 + 4 \sqrt{5}}{1 + \sqrt{2}})$

4 $\sqrt{5} - \sqrt{2} + \log_{e} (\frac{7 + 4 \sqrt{5}}{1 + \sqrt{2}})$