what is the integral of e^2x +1?

Question

if the answer is 1/2e^2x + x ; could u explain why it is not e^2x + x instead of the previous since the integral of e^2x is e^2x not 1/2e^2x.

Answer 1

Actually, the integral of e^(2x) is indeed (1/2)e^(2x), not e^2x. Note that the derivative of e^(2x) is 2e^(2x), not e^(2x), due to the chain rule. The derivative of (1/2)e^(2x), on the other hand, is 2(1/2)e^(2x) = e^(2x).

Thus, the answer is in fact (1/2)e^(2x) + x, plus a constant of course.

Answer 2

You have to use u-substitution. It's like the chain rule of integration.

e^2x +1

let u = 2x
du = 2 dx
dx = (1/2) du

int (e^2x + 1 dx)
= int(e^2x dx) + int(dx)
= (1/2) int(e^u du) + x + C
= 1/2 e^u + x + C
= 1/2 e^2x + x + C

Answer 3

Set y = e^x; note that dy/dx = e^x = y, so dy = y dx

Then integral ( y^2 + 1) dx
= integral (y^2 dx) + integral( 1 dx)
= inegral ( y * y * dx) + x + C
= integral (y dy) + x + C
= (1/2) y^2 + x + C
= (1/2) e^2x + x + C