I'm reading this as

with

.
The value of the integral will be independent of the path if we can find a function

that satisfies the gradient equation above.
You have

Integrate

with respect to

. You get


Differentiate with respect to

. You get
![\dfrac{\partial f}{\partial y}=\dfrac{\partial}{\partial y}[x^2e^{-y}+g(y)]](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cpartial%20f%7D%7B%5Cpartial%20y%7D%3D%5Cdfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%7D%5Bx%5E2e%5E%7B-y%7D%2Bg%28y%29%5D)


Integrate both sides with respect to

to arrive at



So you have

The gradient is continuous for all

, so the fundamental theorem of calculus applies, and so the value of the integral, regardless of the path taken, is
Answer:
x¹²,8x⁴,6x,2x,1
Step-by-step explanation:
x²×x⁶×x⁴=x¹²
x^4(8) =8x⁴
x^1(6) =6x
x^1(2) =2x
x^0 = 1
Answer:
x = 1 or x = -1
Step-by-step explanation:
Given equation:

Factor out -1:

Divide both sides by -1:

Rearrange the terms:



Factor the first two terms and the last two terms separately:

Factor out the common term
:

<u>Zero Product Property</u>: If a ⋅ b = 0 then either a = 0 or b = 0 (or both).
Using the <u>Zero Product Property</u>, set each factor equal to zero and solve for x (if possible):

Therefore, the solutions to the given equation are: x = 1 or x = -1
Learn more here:
brainly.com/question/27751281
brainly.com/question/21186424
1. x+½
2. 82-x
3. 4(x+3/10)
4. 2x-21
5.2x²
6. ⅓(x+53)
Answer:
H = 20.5 hours
I = 19.5 Hours
Step-by-step explanation:
Start by creating a system.
I = time which the inlet pipe takes alone to fill the pond
H = time which the hose takes alone to fill the pond
10/I + 10/H = 10
H - I = 1
H = 20.5 hours
I = 19.5 Hours