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Integrals

Multiple Choice Questions

351.

$\int \cos^{3} (x) . e^{\log (\sin (x))} dx$ is equal to

$- \frac{\sin^{4} (x)}{4} + c$
$- \frac{\cos^{4} (x)}{4} + c$
$\frac{e^{\sin (x)}}{4} + c$
None of the above

352.

The value of $\int_{0}^{\frac{π}{2}} \frac{\cos (3 x) + 1}{2 \cos (x) - 1} dx$ is

2
1
$\frac{1}{2}$
0

353.

The value of $\int_{0}^{1} \tan^{- 1} (\frac{2 x - 1}{1 + x - x^{2}}) dx$ is

1
0
- 1
None of the above

354.

By the application of Simpson's one - third rule for numerical integration, with two subintervals, the value of $\int_{0}^{1} \frac{dx}{1 + x}$ is

$\frac{17}{36}$
$\frac{17}{25}$
$\frac{25}{36}$
$\frac{17}{24}$

355.

$\int \frac{x^{e - 1} + e^{x - 1}}{x^{e} + e^{x}} dx$ is equal to

$\log (x^{e} + e^{x}) + c$
$elog (x^{e} + e^{x}) + c$
$\frac{1}{e} \log (x^{e} + e^{x}) + c$
None of the above

356.
The value of $\int_{\frac{π}{4}}^{\frac{π}{2}} e^{x} (\log (\sin (x)) + \cot (x)) dx$ is

$e^{\frac{π}{4}} \log (2)$

$- e^{\frac{π}{4}} \log (2)$

$\frac{1}{2} e^{\frac{π}{4}} \log (2)$

$- \frac{1}{2} e^{\frac{π}{4}} \log (2)$

$\frac{1}{2} e^{\frac{π}{4}} \log (2)$

$Let I = \int_{\frac{π}{4}}^{\frac{π}{2}} e^{x} (\log (\sin (x)) + \cot (x)) dx \Rightarrow I = \int_{\frac{π}{4}}^{\frac{π}{2}} + \int_{\frac{π}{4}}^{\frac{π}{2}} = \int_{\frac{π}{4}}^{\frac{π}{2}} + - \int_{\frac{π}{4}}^{\frac{π}{2}} = e^{\frac{π}{2}} \log (\sin (\frac{π}{2})) - e^{\frac{π}{4}} \log (\sin (\frac{π}{4})) = e^{- \frac{π}{2}} \log (\frac{1}{\sqrt{2}}) = \frac{1}{2} e^{\frac{π}{4}} \log (2)$