antiderivative question

Question

find

∫ (3^x - 5)(4^x + 7) dx

Answer 1

Hi Ayato,

First you would expand out the integrand (the terms inside the integral sign)

Then you would use the power rule to get the antiderivatives: Int( x^n dx) = (x^{n+1})/n

For example, Int(x^2 dx) = x^3/3.

Hope that helps.

If you could, Im a new tutor here so if you could give me a thumbs up, that would be great! Thank you!

Answer 2

probably the first step is to convert 3^x and 4^x both to powers of e. You can do this by saying e^kx = 3^x and e^kx = 4^x respectively. Do a little algebra on this and see if you can get those bases of 3 and 4 into powers of e like this:

e^kx = 3^x

(ek)x = 3x

ek = 3

k = ln(3)

now you can say e^ln(3)x is the same as 3^x and the term 4^x will work the same way so you can write it as e^ln(4)x

then you can write your integral as:

∫(e^ln(3)x - 4)(e^ln(4)x + 7)dx which can be FOIL'd with distribution pretty easily, then you'll get

∫(e^(ln3+ln4)x - 4e^xln4 + 7e^xln3 - 28)dx. Simplifying the first term even further because ln3+ln4 = ln12, you could write:

∫(e^xln12 - 4e^xln4 + 7e^xln3 - 28)dx. Now you have an integral in which all the individual pieces are easy to integrate. the pieces with powers of e can be solved with u-substitution.