Answer:
Step-by-step explanation:
Student who don't play sports = 24 - 10 = 14
the ratio of students who play sports to those who do not = 10 : 14 = 5:7
Answer:
No your a good person with a kind heart and everyone should think the same
Step-by-step explanation:
Why because all people make mistakes but just because we make a mistakes doenst mean the world will end
Answer:
(x,y) becomes (-y,x) so (3,2) would become (-2,3)
$75.77 × 0.25 = $18.94
The correct answer is B.
Answer:
![\frac{x}{8}-\frac{\sin(4x)}{32}+C](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B8%7D-%5Cfrac%7B%5Csin%284x%29%7D%7B32%7D%2BC)
Step-by-step explanation:
[Most of the work here comes from manipulating the trig to make the term (integrand) integrable.]
Recall that we can express the squared trig functions in terms of cos(2x). That is,
![\cos(2x)=2\cos^2x-1 \\ \cos(2x)=1 - 2\sin^2x.](https://tex.z-dn.net/?f=%5Ccos%282x%29%3D2%5Ccos%5E2x-1%20%5C%5C%20%5Ccos%282x%29%3D1%20-%202%5Csin%5E2x.)
And so inverting these,
.
Multiply them together to obtain an equivalent expression for sin^2(x)cos^2(x) in terms of cos(2x).
![\sin^2x \cdot \cos^2x =\frac{1}{2} (1-\cos2x) \cdot \frac{1}{2} (1+\cos2x) = \frac{1}{4}(1-\cos^2(2x)).](https://tex.z-dn.net/?f=%5Csin%5E2x%20%5Ccdot%20%5Ccos%5E2x%20%3D%5Cfrac%7B1%7D%7B2%7D%20%281-%5Ccos2x%29%20%5Ccdot%20%5Cfrac%7B1%7D%7B2%7D%20%281%2B%5Ccos2x%29%20%3D%20%5Cfrac%7B1%7D%7B4%7D%281-%5Ccos%5E2%282x%29%29.)
Notice we have cos^2(2x) in the integrand now. We've made it worse! Let's try plugging back in to the first identity for cos^2(2x).
![\cos(2x)=2\cos^2x-1 \Rightarrow \cos(4x)=2\cos^2(2x)-1 \Rightarrow \cos^2(2x) = \frac{1}{2}(1+\cos(4x))](https://tex.z-dn.net/?f=%5Ccos%282x%29%3D2%5Ccos%5E2x-1%20%5CRightarrow%20%5Ccos%284x%29%3D2%5Ccos%5E2%282x%29-1%20%5CRightarrow%20%5Ccos%5E2%282x%29%20%3D%20%5Cfrac%7B1%7D%7B2%7D%281%2B%5Ccos%284x%29%29)
So then,
![\sin^2x \cdot \cos^2x = \frac{1}{4}(1-\cos^2(2x)) = \frac{1}{4}(1-\frac{1}{2}(1+\cos(4x))) = \frac{1}{4}(1-\frac{1}{2}-\frac{1}{2}\cos(4x))=\frac{1}{8}(1-\cos(4x)).](https://tex.z-dn.net/?f=%5Csin%5E2x%20%5Ccdot%20%5Ccos%5E2x%20%3D%20%5Cfrac%7B1%7D%7B4%7D%281-%5Ccos%5E2%282x%29%29%20%3D%20%5Cfrac%7B1%7D%7B4%7D%281-%5Cfrac%7B1%7D%7B2%7D%281%2B%5Ccos%284x%29%29%29%20%3D%20%5Cfrac%7B1%7D%7B4%7D%281-%5Cfrac%7B1%7D%7B2%7D-%5Cfrac%7B1%7D%7B2%7D%5Ccos%284x%29%29%3D%5Cfrac%7B1%7D%7B8%7D%281-%5Ccos%284x%29%29.)
This is now integrable (phew),
![\int \sin^2x\cos^2x \ dx = \int \frac{1}{8}(1-\cos(4x)) \ dx = \frac{1}{8} \int (1-\cos(4x)) \ dx = \frac{1}{8}(x-\frac{1}{4}\sin(4x))+C.](https://tex.z-dn.net/?f=%5Cint%20%5Csin%5E2x%5Ccos%5E2x%20%5C%20dx%20%3D%20%5Cint%20%5Cfrac%7B1%7D%7B8%7D%281-%5Ccos%284x%29%29%20%5C%20dx%20%3D%20%5Cfrac%7B1%7D%7B8%7D%20%5Cint%20%281-%5Ccos%284x%29%29%20%5C%20dx%20%3D%20%5Cfrac%7B1%7D%7B8%7D%28x-%5Cfrac%7B1%7D%7B4%7D%5Csin%284x%29%29%2BC.)