First slide

Methods of integration

Question

Let $P (x)$ be a polynomial of least degree whose graph has three points of inflextion (–1, –1), (1,1) and a point with abscissa 0 at which the curve is inclined to the axis of abscissas at an angle of $60^{\circ}$

Difficult

A

1
B

$\sqrt{2} - 1$
C

$2 \sqrt{2} - 1$
D

none of these

Solution

we have $P^{''} (x) = a (x + 1) x (x - 1)$
$\begin{array}{l} = a (x^{3} - x) \\ \Rightarrow P^{'} (x) = a (\frac{1}{4} x^{4} - \frac{1}{2} x^{2}) + b \\ \Rightarrow P (x) = a (\frac{1}{20} x^{5} - \frac{1}{6} x^{3}) + b x + c \end{array}$
As
$\begin{array}{l} - 1 = a (- \frac{1}{20} + \frac{1}{6}) - b + c \\ 1 = a (\frac{1}{20} - \frac{1}{6}) + b + c \\ \Rightarrow c = 0 \end{array}$
and $1 = - \frac{7}{60} a + b$
Also $\sqrt{3} = b$
$\Rightarrow a = 60 (\sqrt{3} - 1) / 7$
we have
$\int_{0}^{1} P (x) d x = - \frac{1}{30} a + \frac{1}{2} b$
$= \frac{2}{7} (1 - \sqrt{3}) + \frac{\sqrt{3}}{2}$

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