![\displaystyle\int\frac{x^3}{\sqrt{x^2+49}}\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7Bx%5E3%7D%7B%5Csqrt%7Bx%5E2%2B49%7D%7D%5C%2C%5Cmathrm%20dx)
Taking
![x=7\tan\theta](https://tex.z-dn.net/?f=x%3D7%5Ctan%5Ctheta)
gives
![\mathrm dx=7\sec^2\theta\,\mathrm d\theta](https://tex.z-dn.net/?f=%5Cmathrm%20dx%3D7%5Csec%5E2%5Ctheta%5C%2C%5Cmathrm%20d%5Ctheta)
, so that the integral becomes
![\displaystyle\int\frac{(7\tan\theta)^3}{\sqrt{(7\tan\theta)^2+49}}(7\sec^2\theta)\,\mathrm d\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%287%5Ctan%5Ctheta%29%5E3%7D%7B%5Csqrt%7B%287%5Ctan%5Ctheta%29%5E2%2B49%7D%7D%287%5Csec%5E2%5Ctheta%29%5C%2C%5Cmathrm%20d%5Ctheta)
![=\displaystyle7^4\int\frac{\tan^3\theta\sec^3\theta}{\sqrt{49\tan^2\theta+49}}\,\mathrm d\theta](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle7%5E4%5Cint%5Cfrac%7B%5Ctan%5E3%5Ctheta%5Csec%5E3%5Ctheta%7D%7B%5Csqrt%7B49%5Ctan%5E2%5Ctheta%2B49%7D%7D%5C%2C%5Cmathrm%20d%5Ctheta)
![=\displaystyle7^3\int\frac{\tan^3\theta\sec^3\theta}{\sqrt{\tan^2\theta+1}}\,\mathrm d\theta](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle7%5E3%5Cint%5Cfrac%7B%5Ctan%5E3%5Ctheta%5Csec%5E3%5Ctheta%7D%7B%5Csqrt%7B%5Ctan%5E2%5Ctheta%2B1%7D%7D%5C%2C%5Cmathrm%20d%5Ctheta)
![=\displaystyle7^3\int\frac{\tan^3\theta\sec^3\theta}{\sqrt{\sec^2\theta}}\,\mathrm d\theta](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle7%5E3%5Cint%5Cfrac%7B%5Ctan%5E3%5Ctheta%5Csec%5E3%5Ctheta%7D%7B%5Csqrt%7B%5Csec%5E2%5Ctheta%7D%7D%5C%2C%5Cmathrm%20d%5Ctheta)
![=\displaystyle7^3\int\frac{\tan^3\theta\sec^3\theta}{|\sec\theta|}\,\mathrm d\theta](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle7%5E3%5Cint%5Cfrac%7B%5Ctan%5E3%5Ctheta%5Csec%5E3%5Ctheta%7D%7B%7C%5Csec%5Ctheta%7C%7D%5C%2C%5Cmathrm%20d%5Ctheta)
When
![\sec\theta>0](https://tex.z-dn.net/?f=%5Csec%5Ctheta%3E0)
, we have
![=\displaystyle7^3\int\frac{\tan^3\theta\sec^3\theta}{\sec\theta}\,\mathrm d\theta](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle7%5E3%5Cint%5Cfrac%7B%5Ctan%5E3%5Ctheta%5Csec%5E3%5Ctheta%7D%7B%5Csec%5Ctheta%7D%5C%2C%5Cmathrm%20d%5Ctheta)
![=\displaystyle7^3\int\tan^3\theta\sec^2\theta\,\mathrm d\theta](https://tex.z-dn.net/?f=%3D%5Cdisplaystyle7%5E3%5Cint%5Ctan%5E3%5Ctheta%5Csec%5E2%5Ctheta%5C%2C%5Cmathrm%20d%5Ctheta)
and from here we can substitute
![u=\tan\theta](https://tex.z-dn.net/?f=u%3D%5Ctan%5Ctheta)
to proceed from here.
Quick note: When we set
![x=7\tan\theta](https://tex.z-dn.net/?f=x%3D7%5Ctan%5Ctheta)
, we are implicitly enforcing
![-\dfrac\pi2](https://tex.z-dn.net/?f=-%5Cdfrac%5Cpi2%3C%5Ctheta%3C%5Cdfrac%5Cpi2)
just so that the substitution can be undone later via
![\theta=\tan^{-1}\dfrac x7](https://tex.z-dn.net/?f=%5Ctheta%3D%5Ctan%5E%7B-1%7D%5Cdfrac%20x7)
. But note that over this domain, we automatically guarantee that
![\sec\theta>0](https://tex.z-dn.net/?f=%5Csec%5Ctheta%3E0)
, so the absolute value bars can be dropped immediately.
Answer: 30 minutes
How: 45 minutes divided by 15 math problems equal 3 minutes per math problem. 3 minutes per math problem times 10 math problems equals 30 minutes on 10 math problems.
45/15=3
3*10=30
The Answer to this problem is (the line y=x). I'm positive that this is the right answer!!!
28 - 2.2x = 11.6x - 54.8
28 + 54.8 = 11.6x + 2.2x
82.8 = 13.8x
82.8 / 13.8 = x
6 = x
or x = 6