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Integrals

Multiple Choice Questions

1001.

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

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

1002.

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

1003.

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

$(x - 1, \sqrt{x})$
$(x - 1, - \sqrt{x})$
$(x + 1, \sqrt{x})$
$(x + 1, - \sqrt{x})$

1004.

$If the value of the integral \int_{0}^{\frac{1}{2}} \frac{x^{2}}{{(1 - x^{2})}^{\frac{3}{2}}} dx x = \sin (θ); dx = \cos (θ) dθ$

$2 \sqrt{3} + π$
$2 \sqrt{3} - π$
$3 \sqrt{2} + π$
$3 \sqrt{2} - π$

Short Answer Type

1005.
The total number of 3-digit numbers, whose sum of digits is 10, is.......

Let xyz be the three digit number

$x + y + z = 10, x \leq 1, y \leq 0, z \leq 0 x - 1 = t \Rightarrow x = 1 + t \{x - 1 \geq 0 t \geq 0\} t + y + z = 10 - 1 t + y + z = 9, 0 \leq t, z, z \leq 9 Total number of non negative integral solution =^{9 + 3 - 1} C_{3 - 1} =^{11} C_{2} = \frac{11 . 10}{2} = 55 But for t = 9, x = 10, so required number of integers = 55 - 1 = 54$

1006.

$Let S be the set of all integer solutions, (x, y, z), of the system of equations x - 2 y + 5 z = 0 - 2 x + 4 y + z = 0 - 7 x + 14 y + 9 z = 0 Such that 15 < x^{2} + y^{2} + z^{2} < 150 . Then, the number of elements in the set S is equal to .$