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Integrals

Multiple Choice Questions

641.

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

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

642.

Considering four sub-intervals, the value of $\int_{0}^{1} \frac{1}{1 + x} dx$ by Trapezoidal rule, is

0.6870
0.6677
0.6977
0.5970

643.

By Simpson's rule, the value of $\int_{1}^{2} \frac{dx}{x}$ dividing the interval (1, 2) into four equal parts, is

0.6932
0.6753
0.6692
7.1324

644.

The value of $\int xsin (x) \sec^{3} (x) dx$ is

$\frac{1}{2} [\sec^{2} (x) - \tan (x)] + c$
$\frac{1}{2} [{xsec}^{2} (x) - \tan (x)] + c$
$\frac{1}{2} [{xsec}^{2} (x) + \tan (x)] + c$
$\frac{1}{2} [\sec^{2} (x) + \tan (x)] + c$

645.

The value of $\int_{0}^{x} {xsin}^{3} (x) dx$ is

$\frac{4 π}{3}$
$\frac{2 π}{3}$
0
None of these

646.
Which of the following is true ?

$\int_{0}^{1} e^{x} dx = e$

$\int_{0}^{1} 2^{x} dx = \log (2)$

$\int_{0}^{1} \sqrt{x} dx = \frac{2}{3}$

$\int_{0}^{1} xdx = \frac{1}{3}$

$\int_{0}^{1} \sqrt{x} dx = \frac{2}{3}$

$(a) \int_{0}^{1} e^{x} dx = {[e^{x}]}_{0}^{1} = e - 1 (b) \int_{0}^{1} 2^{x} dx = {[\frac{2^{x}}{\log_{e} (2)}]}_{0}^{1} = \frac{1}{\log (2)} . (2 - 2^{0}) = \frac{1}{\log (2)} (c) \int_{0}^{1} \sqrt{x} dx = {[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}]}_{0}^{1} = \frac{2}{3} (d) \int_{0}^{1} xdx = {[\frac{x^{2}}{2}]}_{0}^{1} = \frac{1}{2} So, option (c) is correct .$