I can't remember how to integrate ye^ey with respect to y. Is it (y^2)e^xy? or do i need to do integration by parts? i just cant remember.
oops, that was supposed to be ye^xy
Yes, you can do integration by parts. Another way is to integrate e^(xy) and then differentiate the answer w.r.t. x.
Integral of e^(x y)dy = 1/x e^(xy) + c
Differentiate both sides w.r.t. x:
Integral of ye^(x y)dy =
(y/x - 1/x^2) e^(xy) + c'
If you use this method then the integration constant is some arbitrary function of x, so when you differentiate you don't get rid of the integration constant.
To integrate the function ye^ey with respect to y, we can use integration by parts. The formula for integration by parts is:
∫ u dv = uv - ∫ v du
Let's apply this method:
First, we need to determine which part of the function will be u and which part will be dv. In this case, let's choose u = y and dv = e^ey dy.
Next, we differentiate u with respect to y to find du/dy, and we integrate dv to find v:
du/dy = 1
∫ dv = ∫ e^ey dy
To integrate ∫ e^ey dy, we can use a substitution by letting z = ey:
dz/dy = e^ey
dy = dz/e^ey = dz/z
Therefore, the integral becomes:
∫ e^ey dy = ∫ e^z dz/z
Now, we can solve this integral using a logarithmic identity:
∫ e^z dz/z = ln|z| + C
Remembering that z = ey, we substitute back:
∫ e^ey dy = ln|ey| + C
Since ln|ey| is equivalent to y, we have:
∫ ye^ey dy = y + C
So, the correct answer is ye^ey. There is no need for any further integration by parts.