Question:
The integral $\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^{2}+5}}{\sqrt{4 x^{2}-4 x+6}} d x$ is equal to (where $c$ is a constant of integration)
Correct Option: 1
Solution:
$\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^{2}+5}}{\sqrt{(2 x-1)^{2}+5}} d x$
$(2 x-1)^{2}+5=t^{2}$
$2(2 x-1) 2 d x=2 t d t$
$2 \sqrt{t^{2}-5} d x=t d t$
So $\int \frac{\sqrt{t^{2}-5} \cos t}{2 \sqrt{t^{2}-5}} d t=\frac{1}{2} \sin t+c$
$=\frac{1}{2} \sin \sqrt{(2 x-1)^{2}+5}+c$
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