How do you integrate by parts (x*e^(4x)) dx?

1 Answer
Stefan V.
Aug 29, 2015

int(x * e^(4x)dx) = 1/16 * e^(4x) * (4x-1) + c

Explanation:

Remember that the formula that allows you to integrate a function by parts looks like this

color(blue)(int(u * dv) = u * v - int(v * du))" ", where

u, v - are two functions of x;
du, dv - their derivatives.

So, you need to identify u and dv, then calculate du and v.

If you take u = x and dv = e^(4x), you will have

u = x implies du = dx

and

dv = e^(4x) implies v = int(e^(4x)dx) = 1/4 * e^(4x)

Your target integral will thus be

int(x * e^(4x)dx) = x * 1/4 * e^(4x) - int(1/4 e^(4x) * dx)

int(x * e^(4x)dx) = 1/4 * x * e^(4x) - 1/4 * 1/4 * e^(4x) + c

int(x * e^(4x)dx) = 1/4e^(4x)(x - 1/4) + c

int(x * e^(4x)dx) = color(green)(1/16 * e^(4x) * (4x-1) + c)

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