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

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

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

\int \frac{x\,dx}{(\sqrt{7x - 10 - x^2})^3}

can be evaluated by substituting for

x

as (a)

x = \frac{5 + 2t^2}{t^2 + 1}

(b)

x = \frac{5 - t^2}{t^2 + 2}

(c)

x = \frac{2t^2 - 5}{3t^2 - 1}

(d) none of these