![\displaystyle\int\frac{\mathrm dx}{\sin3x\tan3x}=\int\frac{\cos3x}{\sin^23x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dx%7D%7B%5Csin3x%5Ctan3x%7D%3D%5Cint%5Cfrac%7B%5Ccos3x%7D%7B%5Csin%5E23x%7D)
Set
![u=\sin3x](https://tex.z-dn.net/?f=u%3D%5Csin3x)
, so that
![\mathrm du=3\cos3x\,\mathrm dx](https://tex.z-dn.net/?f=%5Cmathrm%20du%3D3%5Ccos3x%5C%2C%5Cmathrm%20dx)
. Then the integral is
![\displaystyle\frac13\int\frac{\mathrm du}{u^2}=-\frac1{3u}+C=-\frac1{3\sin3x}+C=-\frac13\csc3x+C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac13%5Cint%5Cfrac%7B%5Cmathrm%20du%7D%7Bu%5E2%7D%3D-%5Cfrac1%7B3u%7D%2BC%3D-%5Cfrac1%7B3%5Csin3x%7D%2BC%3D-%5Cfrac13%5Ccsc3x%2BC)
Or, if you meant exponents in place of coefficients,
![\displaystyle\int\frac{\mathrm dx}{\sin^3x\tan^3x}=\int\csc^3x\cot^3x\,\mathrm dx=\int\csc^2x\cot^2x\csc x\cot x\,\mathrm dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dx%7D%7B%5Csin%5E3x%5Ctan%5E3x%7D%3D%5Cint%5Ccsc%5E3x%5Ccot%5E3x%5C%2C%5Cmathrm%20dx%3D%5Cint%5Ccsc%5E2x%5Ccot%5E2x%5Ccsc%20x%5Ccot%20x%5C%2C%5Cmathrm%20dx)
Let
![u=\csc x](https://tex.z-dn.net/?f=u%3D%5Ccsc%20x)
, so that
![\mathrm du=-\csc x\cot x\,\mathrm dx](https://tex.z-dn.net/?f=%5Cmathrm%20du%3D-%5Ccsc%20x%5Ccot%20x%5C%2C%5Cmathrm%20dx)
, and use the fact that
![\csc^2x=\cot^2x+1](https://tex.z-dn.net/?f=%5Ccsc%5E2x%3D%5Ccot%5E2x%2B1)
to get
![\displaystyle-\int u^2(u^2-1)\,\mathrm du=\int (u^2-u^4)\,\mathrm du=\frac13u^3-\frac15u^5+C](https://tex.z-dn.net/?f=%5Cdisplaystyle-%5Cint%20u%5E2%28u%5E2-1%29%5C%2C%5Cmathrm%20du%3D%5Cint%20%28u%5E2-u%5E4%29%5C%2C%5Cmathrm%20du%3D%5Cfrac13u%5E3-%5Cfrac15u%5E5%2BC)
Answer:
58 minutes
Step-by-step explanation:
Kelvin and Ornella completed the course in 1 hour 52 minutes
60 minutes + 52 minutes
= 112 minutes
Kelvin time is 54 minutes
Kelvin time is 4 minutes less than ornella
Hence Ornella time is 58 minutes
To write 3.40 as a fraction you have to write 3.40 as numerator and put 1 as the denominator. Now you multiply numerator and denominator by 10 as long as you get in numerator the whole number.
3.40 = 3.40/1 = 34/10
And finally we have:
3.40 as a fraction equals 34/10 simplified we get: 3 2/5
Answer:
4, 2, 8
Step-by-step explanation:
8
the keyword product indicates that it's multiplication and 4 times 2 is 8 and 8 times two is 16