Answer:
A: 1/5
B: 3/5
Step-by-step explanation:
<span>31/50 by the way I love how some people realize so many people answered this question already and decide they should add something that is totally not necessary.
</span>
![\displaystyle\int\sin^3t\cos^3t\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Csin%5E3t%5Ccos%5E3t%5C%2C%5Cmathrm%20dt)
One thing you could do is to expand either a factor of
![\sin^2t](https://tex.z-dn.net/?f=%5Csin%5E2t)
or
![\cos^2t](https://tex.z-dn.net/?f=%5Ccos%5E2t)
, then expand the integrand. I'll do the first.
You have
![\sin^2t=1-\cos^2t](https://tex.z-dn.net/?f=%5Csin%5E2t%3D1-%5Ccos%5E2t)
which means the integral is equivalent to
![\displaystyle\int\sin t(1-\cos^2t)\cos^3t\,\mathrm dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Csin%20t%281-%5Ccos%5E2t%29%5Ccos%5E3t%5C%2C%5Cmathrm%20dt)
Substitute
![u=\cos t](https://tex.z-dn.net/?f=u%3D%5Ccos%20t)
, so that
![\mathrm du=-\sin t\,\mathrm dt](https://tex.z-dn.net/?f=%5Cmathrm%20du%3D-%5Csin%20t%5C%2C%5Cmathrm%20dt)
. This makes it so that the integral above can be rewritten in terms of
![u](https://tex.z-dn.net/?f=u)
as
![\displaystyle-\int(1-u^2)u^3\,\mathrm du=\int(u^5-u^3)\,\mathrm du](https://tex.z-dn.net/?f=%5Cdisplaystyle-%5Cint%281-u%5E2%29u%5E3%5C%2C%5Cmathrm%20du%3D%5Cint%28u%5E5-u%5E3%29%5C%2C%5Cmathrm%20du)
Now just use the power rule:
![\displaystyle\int(u^5-u^3)\,\mathrm du=\dfrac16u^6-\dfrac14u^4+C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%28u%5E5-u%5E3%29%5C%2C%5Cmathrm%20du%3D%5Cdfrac16u%5E6-%5Cdfrac14u%5E4%2BC)
Back-substitute to get the antiderivative back in terms of
![t](https://tex.z-dn.net/?f=t)
:
5 represents one of the dimensions of the rectangle, either length or width.
<h3>
Answer: Choice B</h3>
==========================================
Explanation:
Here's a slightly more detailed breakdown of what's going on
3x + 2y = 15 ... start with the first equation
3x + 2( y ) = 15 .... the parenthesis don't change the expression
3x + 2( 3x+5) = 15 ... replace y with 3x+5; since y = 3x+5
From here, you solve for x like you would any other single variable equation. Once you know x, you can determine y through y = 3x+5.