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Integrals

Multiple Choice Questions

921.

$If f (x) = \frac{1}{x^{2}} \int_{3}^{x} (2 t - 3 f' (t)) dt, then f' (t), then f' (3) is equal to$

$\frac{- 1}{2}$
$\frac{- 1}{3}$
$\frac{1}{2}$
$\frac{1}{3}$

922.

$\int \frac{dx}{(x + 100) \sqrt{x + 99}} = f (x) + c \Rightarrow f (x)$

2(x + 100)^1/2
3(x + 100)^1/2
$2 \tan^{- 1} (\sqrt{x + 99})$
$2 \tan^{- 1} (\sqrt{x + 100})$

923.

$\int \frac{3 - x^{2}}{1 - 2 x + x^{2}} e^{x} dx = e^{x} f (x) + c \Rightarrow f (x)$

$\frac{1 + x}{1 - x}$
$\frac{1 - x}{1 + x}$
$\frac{1 - x}{x - 1}$
$\frac{x - 1}{1 + x}$

924.

$\int \frac{\sqrt{\cot (x)}}{\sin (x) \cos (x)} dx = - f (x) + c \Rightarrow f (x)$

$2 \sqrt{\tan (x)}$
$- 2 \sqrt{\tan (x)}$
$- 2 \sqrt{\cot (x)}$
$2 \sqrt{\cot (x)}$

925.

$\int_{\frac{- π}{2}}^{\frac{π}{2}} \log (\frac{2 - \sin (θ)}{2 + \sin (θ)}) dθ$ is equal to

926.

$\int_{0}^{2} \frac{2 x - 2}{2 x - x^{2}} dx$ is equal to

927.
If $\int \frac{\sin (x)}{\cos (x) (1 + \cos (x))} dx$ = f(x) + c, then f(x) is equal to

$\log |\frac{1 + \cos (x)}{\cos (x)}|$

$\log |\frac{\cos (x)}{1 + \cos (x)}|$

$\log |\frac{\sin (x)}{1 + \sin (x)}|$

$\log |\frac{1 + \sin (x)}{\sin (x)}|$

$\log |\frac{1 + \cos (x)}{\cos (x)}|$

$Let I = \int \frac{\sin (x)}{\cos (x) (1 + \cos (x))} dx Put \cos (x) = t \Rightarrow - \sin (x) dx = dt \Rightarrow I = \int \frac{- dt}{t (1 + t)} = - \int [\frac{1}{t} - \frac{1}{(1 + t)}] dt = - [\log (t) - \log (1 + t)] + c = \log (\frac{t + 1}{t}) + c But I = \int f (x) + c ∴ \log (\frac{1 + \cos (x)}{\cos (x)}) + c = f (x) + c \Rightarrow f (x) = \log |\frac{1 + \cos (x)}{\cos (x)}|$