∫dxx - x is equal to from Mathematics Integra

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Integrals

Multiple Choice Questions

191.

$\int \sqrt{\frac{1 - x}{1 + x}} dx$ is equal to

$\sin^{- 1} (x) + \sqrt{1 - x^{2}} + C$
$\sin^{- 1} (x) - 2 \sqrt{1 - x^{2}} + C$
$2 \sin^{- 1} (x) - \sqrt{1 - x^{2}} + C$
$\sin^{- 1} (x) - \sqrt{1 - x^{2}} + C$

192.

$\int \frac{dx}{1 + \tan (x)}$ is equal to

$\frac{1}{2} + \frac{1}{2} \log |\cos (x) + \sin (x)| + C$
$\frac{x}{2} + \frac{1}{2} \log |\cos (x) - \sin (x)| + C$
$\frac{1}{2} + \frac{1}{2} \log |\cos (x) - \sin (x)| + C$
$\frac{x}{2} + \frac{1}{2} \log |\cos (x) + \sin (x)| + C$

193.

If $\int_{0}^{a} \frac{x^{2} - 1}{1 - x} dx = - \frac{1}{2}$ , then the value of a is equal to

- 1
1
2
- 2

194.

The value of the integral $\int_{0}^{1} x {(1 - x)}^{5} dx$ is equal to

$\frac{1}{6}$
$\frac{1}{7}$
$\frac{6}{7}$
$\frac{1}{42}$

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195.

If [x] denotes the greatest integer less than or equal to x, then the value of $\int_{0}^{2} (|x - 2| + [x]) dx$ is equal to

2
3
1
4

196.

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

$\frac{1}{25} - \frac{6 e^{- 5}}{25}$
$\frac{1}{25} + \frac{6 e^{- 5}}{25}$
$- \frac{1}{25} - \frac{6 e^{- 5}}{25}$
$\frac{1}{25} + \frac{1}{25} e^{- 5}$

197.

$\int \frac{5 x dx}{{(1 - x)}^{3}}$ is equal to

$\frac{5}{2 {(x - 1)}^{2}} - \frac{5}{(x - 1)} + C$
$\frac{5}{2 {(x - 1)}^{2}} + \frac{5}{(x - 1)} + C$
$\frac{5}{3 {(x - 1)}^{2}} + \frac{5}{2 (x - 1)} + C$
$\frac{5}{3 {(x - 1)}^{2}} - \frac{5}{2 (x - 1)} + C$

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198.
$\int \frac{dx}{x - \sqrt{x}}$ is equal to

$2 \log |\sqrt{x} - 1| + C$

$2 \log |\sqrt{x} + 1| + C$

$\log |\sqrt{x} - 1| + C$

$\frac{1}{2} \log |\sqrt{x} + 1| + C$

A.

$2 \log |\sqrt{x} - 1| + C$

$Let I = \int \frac{dx}{x - \sqrt{x}} = \int \frac{dx}{\sqrt{x} (\sqrt{x} - 1)} Put \sqrt{x} - 1 = t \Rightarrow \frac{1}{2 \sqrt{x}} dx = dt ∴ I = \int \frac{2 dt}{t} = 2 \log |t| + C = 2 \log |\sqrt{x} - 1| + C$

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199.

$\int \frac{dx}{4 \sin^{2} (x) + 3 \cos^{2} (x)}$

$\frac{\sqrt{3}}{4} \tan^{- 1} (\frac{2 \tan (x)}{\sqrt{3}}) + C$
$\frac{1}{2 \sqrt{3}} \tan^{- 1} (\frac{\tan (x)}{\sqrt{3}}) + C$
$\frac{2}{\sqrt{3}} \tan^{- 1} (\frac{2 \tan (x)}{\sqrt{3}}) + C$
$\frac{1}{2 \sqrt{3}} \tan^{- 1} (\frac{2 \tan (x)}{\sqrt{3}}) + C$

200.

$\int \frac{\sec (x) dx}{\sqrt{\cos (2 x)}}$ is equal to

2 $\sin^{- 1} (\tan (x))$
$\tan^{- 1} (\frac{\tan (x)}{2}) + C$
$\sin^{- 1} (\tan (x))$
$\frac{\sqrt{3}}{2} \tan^{- 1} (\frac{\tan (x)}{\sqrt{3}}) + C$

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