First slide

Methods of integration

Question

If $f (x) = A \sin \frac{π x}{2} + B, f^{'} (1 / 2) = \sqrt{2}$ and
$\int_{0}^{1} f (x) d x = \frac{2 A}{π}$ then the constants $A$ and $B$ are respectively

Moderate

A

$π / 2, π / 2$
B

$2 / π, 3 / π$
C

$0, - 4 / π$
D

$4 / π, 0$

Solution

$f^{'} (x) = A \frac{π}{2} \cos \frac{π}{2} x \Rightarrow \sqrt{2} = f^{'} (\frac{1}{2}) = \frac{A π}{2 \sqrt{2}}$
so $A = \frac{4}{π}$ Also
$\frac{2 A}{π} = \int_{0}^{1} f (x) d x = - A \frac{2}{π} \cos \frac{π}{2} x + {B x|}_{0}^{1}$
$= B + \frac{2 A}{π} \Rightarrow B = 0$

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