Question

99 POINT QUESTION, AND BRAINLIEST!!!
**Feel free to use any integral, trigonometric, and/or derivative formula(s)...

Evaluate the definite integral from 1 to 14:
⌠1 - cos(4x)
|----------------- dx
⌡4x - sin(4x)


A. (1/4) ln | 56 - sin(56) |               
                 | ---------------- |
                   | 4 - sin(4) |

B. (1/4) ln | 1 - cos(56) |
                | --------------- |
                  | 4 - sin(4) |

C. (1/4) ln | 14 - cos(56) |
                 | ---------------- |
                   | 4 - sin(4) |

D. (1/4) ln | 56 + cos(56) |
               | ------------------- |
                | 4 - sin(4) |

** Sorry for the odd spacing... :(

Please Explain as much as Possible...

Answer 1

$\displaystyle\int_{x=1}^{x=14}\frac{1-\cos4x}{4x-\sin4x}\,\mathrm dx$

First, replace $y=4x$ and $\mathrm du=4\,\mathrm dx$. Then you have

$\displaystyle\frac14\int_{y=4}^{y=56}\frac{1-\cos y}{y-\sin y}\,\mathrm dy$

If we take $z=y-\sin y$, then immediately we see a differential in the numerator in the form of $\mathrm dz=(1-\cos y)\,\mathrm dy$. So the integral becomes

$\displaystyle\frac14\int_{z=4-\sin4}^{z=56-\sin56}\frac{\mathrm dz}z$

and evaluates to

$\dfrac14\ln|z|\bigg|_{z=4-\sin4}^{z=56-\sin56}=\dfrac14\left(\ln|56-\sin56|-\ln|4-\sin4|\right)=\dfrac14\ln\left|\dfrac{56-\sin56}{4-\sin4}\right|$

making (A) the correct answer.