Integrals of the form are calculated with the aid of one of the three Euler substitutions:

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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 function does not appear in the primitive of

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

t

is a function of

x

? (a)

\log_e |t|

(b)

\log_e |t - 2|

(c)

\log_e |t - 1|

(d)

\log |t + 1|