SOLUTION: If f'(x) = e^(2x)sin(x) ... what does f(x) equal? Please explain using antiderividants, integration by parts, and algebraic manipulation.

Algebra ->  Surface-area -> SOLUTION: If f'(x) = e^(2x)sin(x) ... what does f(x) equal? Please explain using antiderividants, integration by parts, and algebraic manipulation.      Log On


   



Question 85524: If f'(x) = e^(2x)sin(x) ... what does f(x) equal? Please explain using antiderividants, integration by parts, and algebraic manipulation.
Answer by Nate(3500) About Me  (Show Source):
You can put this solution on YOUR website!
f'(x) = e^(2x)sin(x)
~
[u = e^(2x)
[du = 2*e^(2x) dx
[dv = sin(x)
[v = - cos(x)
~
int%28+e%5E%282x%29sin%28x%29%2C+dx%2Ca%2Cb%29
-e%5E%282x%29cos%28x%29+-+int%28+-2%2Ae%5E%282x%29%2Acos%28x%29+%2C+dx%2Ca%2Cb%29
-e%5E%282x%29cos%28x%29+%2B+2%2Aint%28+e%5E%282x%29%2Acos%28x%29+%2C+dx%2Ca%2Cb%29
~
[u = e^(2x)
[du = 2*e^(2x) dx
[dv = cos(x)
[v = sin(x)
~
-e%5E%282x%29cos%28x%29+%2B+2%2Aint%28+e%5E%282x%29%2Acos%28x%29+%2C+dx%2Ca%2Cb%29



~
y+=+int%28+e%5E%282x%29sin%28x%29%2C+dx%2Ca%2Cb%29
~

y+=+-e%5E%282x%29cos%28x%29+%2B+2e%5E%282x%29sin%28x%29+-+4y
5y+=+e%5E%282x%29%282%2Asin%28x%29+-+cos%28x%29%29
y+=+%281%2F5%29e%5E%282x%29%282%2Asin%28x%29+-+cos%28x%29%29
~
Constant Rule:
~
y+=+%281%2F5%29e%5E%282x%29%282%2Asin%28x%29+-+cos%28x%29%29+%2B+C