‘Easy’ Integrals

Abstract

You should always be alert, when confronted by a definite integral, for the happy possibility that although the integral might look ‘interesting’ (that is, hard!) just maybe it will still yield to a direct, frontal attack. The first six integrals in this chapter are in that category. If a and b are positive constants, calculate:

$$ {\displaystyle {\int}_1^{\infty}\frac{1}{\left(\mathrm{x}+\mathrm{a}\right)\sqrt{\mathrm{x}-1}}\ \mathrm{dx}} $$

and

$$ {\displaystyle {\int}_0^{\infty } \ln \left(1+\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2}\right)\ \mathrm{dx}} $$

and

$$ {\displaystyle {\int}_0^{\infty}\frac{ \ln \left(\mathrm{x}\right)}{{\mathrm{x}}^2+{\mathrm{b}}^2}\ \mathrm{dx}} $$

and

$$ {\displaystyle {\int}_0^{\infty}\frac{1}{1+{\mathrm{e}}^{\mathrm{ax}}}\ \mathrm{dx}}. $$