The integral (2x-1)(2x-1)^2+5√4x^2-4x+6,dx is equal to (where c is a constant of integrati…

Question

The integral $\displaystyle\int\frac{(2x-1)\cos\sqrt{(2x-1)^2+5}}{\sqrt{4x^2-4x+6}}\,dx$ is equal to (where $c$ is a constant of integration)

Accepted Answer

Note $4x^2-4x+6=(2x-1)^2+5$. Let $z^2=(2x-1)^2+5$, so $2(2x-1)\cdot2\,dx=2z\,dz \implies (2x-1)\,dx=\dfrac12 z\,dz$:

$$I = \int\frac{\cos z}{z}\cdot\frac12 z\,dz = \frac12\int\cos z\,dz = \frac12\sin z+C = \frac12\sin\sqrt{(2x-1)^2+5}+c$$

Solutions · 1