# The integral

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)

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

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

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

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

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 $\int \frac{\sqrt{t^{2}-5} \cos t}{2 \sqrt{t^{2}-5}} d t=\frac{1}{2} \sin t+c$

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