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 $\mathrm{c}$ is a constant of integration)
Correct Option: 1
Solution:
$\int \frac{(2 x-1) \cos \sqrt{(2 x-1)^{2}+5}}{\sqrt{(2 x-1)^{2}+5}} d x$
$(2 \mathrm{x}-1)^{2}+5=\mathrm{t}^{2}$
$2(2 \mathrm{x}-1) 2 \mathrm{dx}=2 \mathrm{t} \mathrm{dt}$
$2 \sqrt{\mathrm{t}^{2}-5} \mathrm{~d} \mathrm{x}=\mathrm{t} \mathrm{dt}$
So $\int \frac{\sqrt{t^{2}-5} \cos t}{2 \sqrt{t^{2}-5}} d t=\frac{1}{2} \sin t+c$
$=\frac{1}{2} \sin \sqrt{(2 x-1)^{2}+5}+c$