First slide

Methods of integration

Question

$\int \frac{(2 x - 1) \cos \sqrt{(2 x - 1)^{2} + 5}}{\sqrt{4 x^{2} - 4 x + 6}} d x is equal to$

Moderate

A

$\frac{1}{2} \sin \sqrt{{(2 x - 1)}^{2} + 5} + c$
B

$\frac{1}{2} \cos \sqrt{{(2 x + 1)}^{2} + 5} + c$
C

$\frac{1}{2} \cos \sqrt{{(2 x - 1)}^{2} + 5} + c$
D

$\frac{1}{2} \sin \sqrt{{(2 x + 1)}^{2} + 5} + c$

Solution

$\begin{array}{l} The given integral becomes \int \frac{(2 x - 1) \cos \sqrt{{(2 x - 1)}^{2} + 5}}{\sqrt{{(2 x - 1)}^{2} + 5}} d x \\ Substitute {(2 x - 1)}^{2} + 5 = t^{2}, and 2 (2 x - 1) .2 d x = 2 t d t \\ \int \frac{\cos (t)}{t} . \frac{1}{2} t d t = \frac{1}{2} \int \cos t d t = \frac{1}{2} \sin t = \frac{1}{2} \sin (\sqrt{{(2 x - 1)}^{2} + 5}) + C \end{array}$

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