SOLUTION: Evaluate the indefinite integral using substitution for: 3^x.

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Question 585836: Evaluate the indefinite integral using substitution for: 3^x.
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
Evaluate the indefinite integral using substitution for: 3^x.

int%283%5Ex%2Cdx%29

Let u = 3%5Ex

ln(u) = ln(3%5Ex)

ln(u) = x·ln(3)

Take the differential of both sides, remembering that ln(3) is a constant:

%28du%29%2Fu = ln(3)·dx

Solve for dx by dividing both sides by ln(3)

%28du%29%2F%28u%2Aln%283%29%29 = dx

Substitute u for 3%5Ex and %28du%29%2F%28u%2Aln%283%29%29 for dx:

int%283%5Ex%2Cdx%29 = int%28u%2Aexpr%28%28du%29%2F%28u%2Aln%283%29%29%29%29 = int%28u%2Adu%2F%28u%2Aln%283%29%29%29 = int%28cross%28u%29%2Adu%2F%28cross%28u%29%2Aln%283%29%29%29 = int%28%28du%29%2Fln%283%29%29 = expr%281%2Fln%283%29%29%2Aint%28du%29 = expr%281%2Fln%283%29%29u + C = expr%281%2Fln%283%29%29u + C = expr%281%2Fln%283%29%293%5Ex + C = 3%5Ex%2Fln%283%29%29 + C

Edwin