SOLUTION: Very confused on taking antiderivatives. In my class I am not allowed to use integrals, just math logical to solve this problem: e^(x/5) + sin(3x). I know it's sort of like figurin

Question 999361: Very confused on taking antiderivatives. In my class I am not allowed to use integrals, just math logical to solve this problem: e^(x/5) + sin(3x). I know it's sort of like figuring out what the function must have been before the derivative was taken but this keeps giving me trouble. I have to reverse the derivative and get it back to its original state. sin(3x) is simple enough for it to be +sin(3x) it must be a negative cosine and something for which cancels the 3 out so 1/3 therefore, -1/3cos(3x) would be the original. But the original of e^(x/5) this one baffles me. It's really the (x/5) that I don't get. How does undoing that work exactly? what must I do/understand first before getting its antiderivative/
Thank you!
Found 2 solutions by Alan3354, stanbon:
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Very confused on taking antiderivatives. In my class I am not allowed to use integrals, just math logical to solve this problem: e^(x/5) + sin(3x). I know it's sort of like figuring out what the function must have been before the derivative was taken but this keeps giving me trouble. I have to reverse the derivative and get it back to its original state. sin(3x) is simple enough for it to be +sin(3x) it must be a negative cosine and something for which cancels the 3 out so 1/3 therefore, -1/3cos(3x) would be the original. But the original of e^(x/5) this one baffles me. It's really the (x/5) that I don't get. How does undoing that work exactly? what must I do/understand first before getting its antiderivative/
========================
d/dx(e^x) = e^x
---
d/dx(e^(ax)) = a*e^(ax)

Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
f'(x) = e^(x/5)dx
Find f(x)
-----
Let x/5 = u
Then du = (1/5)dx
dx = 5du
-----
dy = e^u(5du)
---
dy = 5e^u du
--
Integrate to get:
y = 5e^u
---
Substitute to get:
y = 5e^(x/5)
-------
Cheers,
Stan H.
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