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Integrals

Multiple Choice Questions

711.

$\int \frac{x + \sin (x)}{1 + \cos (x)} dx = ?$

$xtan (\frac{x}{2}) + C$
$xsin (\frac{x}{2}) + \cos (\frac{x}{2}) + C$
$xtan (\frac{x}{2}) + \sec (\frac{x}{2}) + C$
$xtan (\frac{x}{2}) + \sec (\frac{x}{2}) + C$

712.

$If I_{n} = \int \frac{\sin (nx)}{\cos (x)} dx, then I_{n} = ?$

$\frac{- 2}{n - 1} \cos (n - 1) x - I_{n - 2}$
$\frac{2}{n - 1} \cos (n - 1) x + I_{n - 2}$
$\frac{- 2}{n + 1} \sin (n - 1) x - I_{n - 2}$
$\frac{- 2}{n + 1} \cos (n - 1) x - I_{n - 2}$

Short Answer Type

713.

$The integral \int_{0}^{2} ||x - 1| - x| d x = ?$

714.
Let [t] denote the greatest integer less than or equal to t. Then the value of $\int_{1}^{2} |2 x - [3 x]| d x$ is

Ans : 1

$\int_{1}^{2} |3 x - [3 x] - x| d x \Rightarrow \int_{1}^{2} |\{3 x\} - x| d x = \int_{1}^{2} |x - [3 x]| d x \Rightarrow \int_{1}^{2} |x| d x - \int_{1}^{2} |\{3 x\}| d x \Rightarrow {(\frac{x^{2}}{2})}_{1}^{2} - 3 . \int_{0}^{^{\frac{1}{3}}} 3 x d x = (\frac{4}{2} - \frac{1}{2}) - 9 {(\frac{x^{2}}{2})}_{0}^{\frac{1}{3}} \Rightarrow \frac{3}{2} - \frac{9}{2} ({(\frac{1}{3})}^{2} - 0^{2}) = 1$

Multiple Choice Questions

715.

$Area enclosed by 0 \leq y \leq x^{2} + 1, 0 \leq y \leq x + 1, \frac{1}{2} \leq x \leq 2 is :$

$\frac{1}{12}$
$\frac{1}{6}$
1
$\frac{1}{3}$

716.

$Evaluate \int_{- π}^{π} |π - |x|| d x$

$π^{2}$
$\frac{π^{2}}{2}$
$\frac{π^{2}}{3}$
$\frac{π^{2}}{4}$

717.

$(2^{\frac{1}{5}} + 5^{\frac{1}{8}}) has integral terms find least value of n is$

718.

$If \int \sin^{- 1} (\sqrt{\frac{x}{1 + x}}) dx = A (x) \tan^{- 1} (\sqrt{x}) + Bx + C, where C is a constant of integration, then ordered pair (A (x), B (x)) can be :$