First slide

Evaluation of definite integrals

Question

$If f (0) = 1, f (2) = 3, f^{'} (2) = 5, then find the value of \int_{0}^{1} x f^{''} (2 x) d x$

Moderate

A

1
B

2
C

3
D

4

Solution

$I_{1} = \int_{0}^{1} x f^{''} (2 x) d x . Putting t = 2 x, i.e.$
$d x = \frac{d t}{2}, we get I_{1} = \frac{1}{4} \int_{0}^{2} t f^{''} (t) d t$
$\begin{array}{rcl} = & \frac{1}{4} [{t f^{'} (t)|}_{0}^{2} - \int_{0}^{2} f^{'} (t) d t] (Integrating by parts) \end{array}$
$= \frac{1}{4} [{t f^{'} (t)|}_{0}^{2} - {f (t)|}_{0}^{2}]$
$= \frac{1}{4} (2 f^{'} (2) - f (2) + f (0)) = \frac{1}{4} (10 - 3 + 1) = 2$

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