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Integrals

Multiple Choice Questions

811.

The value of $\int \frac{e^{6 \log (x)} - e^{5 \log (x)}}{e^{4 \log (x)} - e^{3 \log (x)}} dx$ is equal to

0 + C
$\frac{x^{3}}{3} + C$
$\frac{3}{x^{3}} + C$
$\frac{1}{x} + C$

812.

$\int_{0}^{\frac{π}{2}} \frac{\sin^{1000} (x)}{\sin^{1000} (x) + \cos^{1000} (x)} dx$ is equal to

1000
1
$\frac{π}{2}$
$\frac{π}{4}$

813.

The vaue of $\int_{2}^{8} \frac{\sqrt{10 - x}}{\sqrt{x} + \sqrt{10 - x}} dx$ is

814.

$\int_{\frac{- π}{2}}^{\frac{π}{2}} \frac{dx}{e^{\sin (x)} + 1}$

1
0
$\frac{π}{2}$
$- \frac{π}{2}$

815.

$\int_{- 5}^{5} |x + 2| dx$ is equal to

816.
$\int_{0}^{\frac{π}{2}} \frac{1}{a^{2} . \sin^{2} (x) + b^{2} . \cos^{2} (x)} dx$

$\frac{π}{2 ab}$

$\frac{πb}{4 a}$

$\frac{πa}{2 b}$

$\frac{πa}{4 b}$

$\frac{π}{2 ab}$

$I = \int_{0}^{\frac{π}{2}} \frac{1}{a^{2} . \sin^{2} (x) + b^{2} . \cos^{2} (x)} dx = \int_{0}^{\frac{π}{2}} \frac{\sec^{2} (x)}{a^{2} . \tan^{2} (x) + b^{2}} dx Put \tan (x) = t \Rightarrow \sec^{2} (x) dx = dt When x = 0, t = \tan (0) and x = \frac{π}{2}, t = \tan (\frac{π}{2}) = \infty ∴ I = \int_{0}^{\infty} \frac{dt}{a^{2} t^{2} + b^{2}} = \frac{1}{a^{2}} \int_{0}^{\infty} \frac{dt}{t^{2} + {(\frac{b}{a})}^{2}} = \frac{1}{a^{2}} . \frac{1}{(\frac{b}{a})} {[\tan^{- 1} (\frac{t}{(\frac{b}{a})})]}_{0}^{\infty} = \frac{1}{ab} [\tan^{- 1} (\infty) - \tan^{- 1} (0)] = \frac{1}{ab} \frac{π}{2} = \frac{π}{2 ab}$

817.

$\int \sqrt{x^{2} + 2 x + 5} dx$ is equal to

$\frac{1}{2} (x + 1) \sqrt{x^{2} + 2 x + 5} + 2 \log |x + 1 + \sqrt{x^{2} + 2 x + 5}| + C$
$(x + 1) \sqrt{x^{2} + 2 x + 5} + \frac{1}{2} \log |x + 1 + \sqrt{x^{2} + 2 x + 5}| + C$
$(x + 1) \sqrt{x^{2} + 2 x + 5} + 2 \log |x + 1 + \sqrt{x^{2} + 2 x + 5}| + C$
$(x + 1) \sqrt{x^{2} + 2 x + 5} - 2 \log |x + 1 + \sqrt{x^{2} + 2 x + 5}| + C$