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Integrals

Multiple Choice Questions

711.

$\int_{0}^{1} \frac{dx}{\sqrt{1 + x} + \sqrt{x}}$ is equal to

$\frac{4}{3} (\sqrt{2} - 1)$
$\frac{3}{4} (\sqrt{2} - 1)$
$\frac{4}{3} (1 - \sqrt{2})$
$\frac{3}{4} (1 - \sqrt{2})$

712.

$\int_{0}^{1} |5 x - 3| dx$ is equal to

$\frac{10}{13}$
$\frac{31}{10}$
$\frac{13}{10}$
None of these

713.

$\int_{0}^{π} {xsin}^{4} (x) dx$ is equal to

$\frac{3 π}{16}$
$\frac{3 π^{2}}{16}$
$\frac{16 π}{3}$
$\frac{16 π^{2}}{3}$

714.

$\int 5^{5^{5^{x}}} 5^{5^{x}} 5^{x} dx$ is equal to

$5^{5^{5^{x}}} {(\log (5))}^{3} + c$
$\frac{5^{5^{5^{x}}}}{{(\log (5))}^{3}} + c$
$\frac{5^{^{5^{x}}}}{{(\log (5))}^{3}} + c$
$5^{^{5^{x}}} {(\log (5))}^{3} + c$

715.

The value of $\int_{2}^{4} \{|x - 2| + |x - 3|\} dx$ is

1 sq unit
2 sq unit
3 sq unit
5 sq unit

716.

If b > a, then $\int_{a}^{b} \frac{dx}{\sqrt{(x - a) (b - x)}}$ is equal to

$\frac{π}{2}$
$π$
$\frac{π}{2} (b - a)$
$\frac{π}{4} (b - a)$

717.
$\int e^{x} \frac{x^{2} + 1}{{(x + 1)}^{2}} dx$ is equal to

$\frac{- e^{x}}{x + 1} + c$

$\frac{e^{x}}{x + 1} + c$

$\frac{e^{x} (x - 1)}{x + 1} + c$

$\frac{{xe}^{x}}{x + 1} + c$

$\frac{e^{x} (x - 1)}{x + 1} + c$

$Let I = \int e^{x} \frac{x^{2} + 1}{{(x + 1)}^{2}} dx = \int e^{x} \frac{(x - 1)}{(x + 1)} dx + 2 \int \frac{e^{x}}{{(x + 1)}^{2}} dx = \int e^{x} dx - 2 \int \frac{e^{x}}{(x + 1)} dx + 2 \int \frac{e^{x}}{{(x + 1)}^{2}} dx = e^{x} - 2 \int \frac{e^{x}}{(x + 1)} dx + 2 [- \frac{e^{x}}{(x + 1)} - \int - \frac{e^{x}}{(x + 1)} dx] = e^{x} - \frac{2 e^{x}}{x + 1} + c = \frac{e^{x} (x - 1)}{x + 1} + c$