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Integrals

Multiple Choice Questions

741.

$\int \frac{dx}{x (x^{7} + 1)}$ is equal to

$\log (\frac{x^{7}}{x^{7} + 1}) + c$
$\frac{1}{7} \log (\frac{x^{7}}{x^{7} + 1}) + c$
$\log (\frac{x^{7} + 1}{x^{7}}) + c$
$\frac{1}{7} \log (\frac{x^{7} + 1}{x^{7}}) + c$

742.

$\int_{- 1}^{1} |1 - x| dx$ is equal to

743.

$\int \sqrt{x} e^{\sqrt{x}} dx$ is equal to

$2 \sqrt{x} - e^{\sqrt{x}} - 4 \sqrt{{xe}^{\sqrt{x}}} + c$
$(2 \sqrt{x} - 4 \sqrt{x} + 4) e^{\sqrt{x}} + c$
$(2 \sqrt{x} + 4 \sqrt{x} + 4) e^{\sqrt{x}} + c$
$(1 - 4 \sqrt{x}) e^{\sqrt{x}} + c$

744.

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

$\sin^{- 1} (x + 1) + c$
$\sinh^{- 1} (x + 1) + c$
$\tanh^{- 1} (x + 1) + c$
$\tan^{- 1} (x + 1) + c$

745.

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

746.
$\int_{0}^{π} \frac{xdx}{a^{2} \cos^{2} (x) + b^{2} \sin^{2} (x)} dx$ is equal to

$\frac{π}{2 ab}$

$\frac{π}{ab}$

$\frac{π^{2}}{2 ab}$

$\frac{π^{2}}{ab}$

$\frac{π^{2}}{ab}$

$Let I = \int_{0}^{π} \frac{xdx}{a^{2} \cos^{2} (x) + b^{2} \sin^{2} (x)} dx . . . (i) I = \int_{0}^{π} \frac{(π - x)}{a^{2} \cos^{2} (π - x) + b^{2} \sin^{2} (π - x)} [Using \int_{0}^{a} f (x) dx = \int_{0}^{a} f (a - x) dx] or I = \int_{0}^{π} \frac{(π - x)}{a^{2} \cos^{2} (x) + b^{2} \sin^{2} (x)} dx . . . (ii) Adding Eqs . (i) and (ii), 2 I = \int_{0}^{π} \frac{x + π - x}{(a^{2} \cos^{2} (x) + b^{2} \sin^{2} (x))} = π \int_{0}^{π} \frac{dx}{a^{2} \cos^{2} (x) + b^{2} \sin^{2} (x)} = 2 π \int_{0}^{\frac{π}{2}} \frac{dx}{a^{2} \cos^{2} (x) + b^{2} \sin^{2} (x)} [∵ \int_{0}^{2 a} f (x) dx = 2 \int_{0}^{a} f (x) dx,] = 2 π \int_{0}^{\frac{π}{2}} \frac{x}{a^{2} + b^{2} \tan^{2} (x)}$

$Now, put \tan x = t ∴ dx = \frac{dt}{\sec^{2} (x)} = 2 π \int_{0}^{\infty} \frac{dt}{a^{2} + b^{2} t^{2}} = \frac{1}{b^{2}} \times 2 π \int_{0}^{\infty} \frac{dt}{\frac{a^{2}}{b^{2}} + t^{2}} = [\frac{2 π}{b^{2}} . \frac{b}{a} \tan^{- 1} (\frac{bt}{a})] = \frac{2 π}{ab} [\tan^{- 1} (\infty) - \tan^{- 1} (0)] = \frac{2 π}{ab} (\frac{π}{2} - 0) = \frac{2 π}{ab} \times \frac{π}{2} = \frac{π^{2}}{ab}$