can be integrated this way:
let
so,
substitute u and dx to get:
and integrate to complete ....
I'm having trouble evaluating thhis integral
S = integral sign
S 4 ( 5^4x+9) dx
I tried using X^n = (X^n+1 / n+1) + C by taking the constant 4 outside and to the left but the result after applying the above rule made no sense.
I then tried using substitution where 4x+9 = alpha. So the derivative of alpha is just 4 but cannot reconstruct what this will give. Here's what I ended up with:
alpha = @= 4x+9
derivative @ = d@ = 4
d@ / 4 = dx
thus,
S 4 (5^4x+9) dx= (5^4x+9) d@/4 but now I don't know what to do.
Your "power rule", that the anti-derivative of is requires that x be the base not the exponent.
The derivative of is and the anti-derivative is (which you can derive by taking logarithms as harish21 did).
To integrate , let u= 4x+ 9 so that du= 4dx, dx= du/4. The integral becomes .