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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

194.

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

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

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

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

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

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

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

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

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

$Let I = \int \frac{5 x}{{(1 - x)}^{3}} Let \frac{5 x}{{(1 - x)}^{3}} = \frac{A}{(1 - x)} + \frac{B}{{(1 - x)}^{2}} + \frac{C}{{(1 - x)}^{3}} 5 x = A {(1 - x)}^{2} + B (1 - x) + C \Rightarrow 5 x = A (1 - 2 x + x^{2}) + B (1 - x) + C On equating the coefficients of x^{2}, x and constant terms, we get 0 = A, 5 = - 2 A - B, 0 = A + b + C ∴ 5 = - 2 (0) - B \Rightarrow B = - 5 and 0 = 0 - 5 + C \Rightarrow C = 5 ∴ \frac{5 x}{{(1 - x)}^{3}} = 0 + \frac{- 5}{{(1 - x)}^{3}} + \frac{5}{{(1 - x)}^{3}}$

On integrating both sides, we get

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