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Standard Formulae - 1

integral from...

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integral from 0 to infinity x/(1+x)(1+x^2) dx

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Q. The value of the integral

\int_{0}^{\infty} \frac{x dx}{(1 + x) (1 + x^{2})}

Q. The number of integral solutions to the system of equations

x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 20

x_{1} + x_{2} = 15

x_{k} \geq 0

Q. How many integral solutions are there to the system of equations

x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 20

x_{1} + x_{2} = 15

x_{k} \geq 0

Q. If x is positive, the sum to infinity of the series

\frac{1}{1 + x} - \frac{1 - x}{(1 + x)^{2}} + \frac{(1 - x)^{2}}{(1 + x)^{3}} - \frac{(1 - x)^{3}}{(1 + x)^{4}} + . . . . . . is

(a) 1/2
(b) 3/4
(c) 1
(d) none of these

x_{1}

x_{2}

be the roots of the equation

2 x^{2} + 6 x + b = 0.

\frac{x_{1}}{x_{2}} + \frac{x_{2}}{x_{1}} = k

b < 0,

then the minimum integral value of

| k |

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