Mathematics

Problem

Integrals of the form

\int \sqrt{ax^2 + bx + c}\,dx

are calculated with the aid of one of the three Euler substitutions: (i)

\sqrt{ax^2 + bx + c} = t \pm x\sqrt{a}

a > 0

(ii)

\sqrt{ax^2 + bx + c} = tx \pm \sqrt{c}

c > 0

(iii)

\sqrt{ax^2 + bx + c} = (x - \alpha)t

ax^2 + bx + c = a(x - \alpha)(x - \beta)

i.e., if

\alpha

is a real root of

ax^2 + bx + c = 0

Which of the following functions does not appear in the primitive of

\frac{1}{1 + \sqrt{x^2 + 2x + 2}}

t

is a function of

x

? (a)

\log_e |t + 1|

(b)

\log_e |t + 2|

(c)

\frac{1}{t + 2}

(d) None of these (a)

\log_e |t + 1|

(b)

\log_e |t + 2|

(c)

\frac{1}{t + 2}

(d) None of these