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Integration by recognition
Hey would it be possible for someone to help me, i've been ill and my teacher won't explain this to me again.
My text book just says that:
The integral of (ax+b)^(n+1) =
1/a(n+1) * (ax + b)^(n+1) + C
I also don't get this part:
Integal of (k(f(x))^n) x f(dash) (x) =
k/(n+1) * (f(x)^(n+1) + C)
Sorry i know it's hard to read but someone help me please!
My text book just says that:
The integral of (ax+b)^(n+1) =
1/a(n+1) * (ax + b)^(n+1) + C
I also don't get this part:
Integal of (k(f(x))^n) x f(dash) (x) =
k/(n+1) * (f(x)^(n+1) + C)
Sorry i know it's hard to read but someone help me please!
For the first one use the substitution u = ax+b, do du = adx (ie dx = du/a ), turning
Int [ (ax+b)n dx ] = (1/a) Int [ un du ] = (1/a) un+1/(n+1) = (1/a) (ax+b)n+1/(n+1)
For the second, let u = f(x), so du = f'(x)dx
Int [ f(x)nf'(x) dx ] = Int [ undu ] = un+1/(n+1) = f(x)n+1/(n+1)
+ constants etc
Int [ (ax+b)n dx ] = (1/a) Int [ un du ] = (1/a) un+1/(n+1) = (1/a) (ax+b)n+1/(n+1)
For the second, let u = f(x), so du = f'(x)dx
Int [ f(x)nf'(x) dx ] = Int [ undu ] = un+1/(n+1) = f(x)n+1/(n+1)
+ constants etc
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