The original equation:
Integral of csc(5x)
Use a u sub:
u = 5x
du = 5dx
Simplify the du:
![\frac{1}{5} du = dx](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B5%7D%20du%20%3D%20dx)
Apply to the equation:
Integral csc(u)
![\frac{1}{5}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B5%7D%20)
du
Simplify:
Integral
![\frac{1}{5}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B5%7D%20)
![\frac{1}{sin(u)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bsin%28u%29%7D%20)
du
Factor out the constant:
![\frac{1}{5}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B5%7D%20)
Integral
![\frac{1}{sin(u)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bsin%28u%29%7D%20)
du
Use a second Substitution:
v = tan(
![\frac{u}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bu%7D%7B2%7D%20)
)
du =
![\frac{2}{1+v^{2} }](https://tex.z-dn.net/?f=%20%5Cfrac%7B2%7D%7B1%2Bv%5E%7B2%7D%20%7D%20)
dv
Applying to the equation:
![\frac{1}{5} Integral \frac{1}{ \frac{2v}{1+ v^{2} } } * \frac{2}{1+v^{2} } dv](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B5%7D%20Integral%20%20%5Cfrac%7B1%7D%7B%20%5Cfrac%7B2v%7D%7B1%2B%20v%5E%7B2%7D%20%7D%20%7D%20%2A%20%5Cfrac%7B2%7D%7B1%2Bv%5E%7B2%7D%20%7D%20dv)
Simplify:
![\frac{1}{5} Integral \frac{1}{v} dv](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B5%7D%20Integral%20%20%5Cfrac%7B1%7D%7Bv%7D%20dv)
Integrate:
![\frac{1}{5} ln(|v|)](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B5%7D%20ln%28%7Cv%7C%29)
Insert back in your v and u:
v = tan(
![\frac{u}{2}](https://tex.z-dn.net/?f=%20%5Cfrac%7Bu%7D%7B2%7D%20)
)
u = 5x
This gives us the final equation (don't forget your constant):
![\frac{1}{5} ln(|tan( \frac{5x}{2} )|) + C](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B5%7D%20ln%28%7Ctan%28%20%5Cfrac%7B5x%7D%7B2%7D%20%29%7C%29%20%2B%20C)
(Thank you for making me write it out, I made a mistake on the original answer.)
Answer:
1.5x=y
Step-by-step explanation:
Answer:
1
Step-by-step explanation:
1 deca=10 liters
I multiply the decaliter amout by ten.
we know that
the verbal expression "all real numbers less than
"
represent the inequality--------> ![x < 69](https://tex.z-dn.net/?f=x%20%3C%2069)
the solution is the interval-------> (-∞,69)
using a graphing tool
see the attached figure
the solution is the shaded area
<u>the answer is</u>
the inequality is ![x < 69](https://tex.z-dn.net/?f=x%20%3C%2069)
Answer:
(6,-3)
Step-by-step explanation:
online calculator