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Integrals

Multiple Choice Questions

781.

If $\int_{0}^{1} f (x) dx = 5, then the value of . . . + \int_{0}^{1} x^{9} f (x^{10}) dx$ is equal to

782.

If $\int f (x) \sin (x) . \cos (x) dx = \frac{1}{2 (b^{2} - a^{2})} \log (f (x)) + c$ , where c is the constant of integration, then f(x) is

$\frac{2}{abcos (2 x)}$
$\frac{2}{(b^{2} - a^{2}) \cos (2 x)}$
$\frac{2}{ab \sin (2 x)}$
$\frac{2}{(b^{2} - a^{2}) \sin (2 x)}$

783.

If $\int \frac{\sqrt{x}}{x (x + 1)} dx = {ktan}^{- 1} (m)$ , then (k, m) is

(2, x)
(1, x)
$(1, \sqrt{x})$
$(2, \sqrt{x})$

784.

$\int_{0}^{\frac{π}{4}} \frac{\sin (x) + \cos (x)}{3 + \sin (2 x)} dx$ is

$\frac{1}{4} \log (3)$
$\log (3)$
$\frac{1}{2 \log (3)}$
$2 \log (3)$

785.

$\int_{0}^{1} x {(1 - x)}^{\frac{3}{2}} dx$ is

$- \frac{2}{35}$
$\frac{4}{35}$
$\frac{24}{35}$
$- \frac{8}{35}$

786.

The value of $\int_{0}^{4} |x - 1| dx$ is

$\frac{5}{2}$
5
4
1

787.

If I_n = $\int_{0}^{\frac{π}{4}} \tan^{n} (x) dx$ , where where n is apositive integer, then $I_{10} + I_{8}$ is

$\frac{1}{9}$
$\frac{1}{8}$
$\frac{1}{7}$
9

788.
If I_n = $\int e^{x} [\frac{\sin (x) + \cos (x)}{1 - \sin^{2} (x)}] dx$ is

$(e^{x} . \csc (x)) + C$

$(e^{x} . \cot (x)) + C$

$(e^{x} . \sec (x)) + C$

$(e^{x} . \tan (x)) + C$

$(e^{x} . \sec (x)) + C$

$\int e^{x} [\frac{\sin (x) + \cos (x)}{1 - \sin^{2} (x)}] dx = \int e^{x} (\frac{\sin (x) + \cos (x)}{\cos^{2} (x)}) dx = \int e^{x} \tan (x) . \sec (x) dx + \int \underset{I}{e^{x}} . \underset{II}{\sec (x)} dx = \int e^{x} \tan (x) . \sec (x) dx + \{\sec (x) . e^{x} - \int \sec (x) . \tan (x) e^{x} dx\} = \int e^{x} \tan (x) . \sec (x) dx + e^{x} \sec (x) - \int e^{x} \tan (x) . \sec (x) dx = (e^{x} . \sec (x)) + C$