Question

evaluate integral of x^3e^-2xdx​

Answer 1

Answer:

114584/*897/*85*/5

Step-by-step explanation:

Answer 2

Would need to use integration by parts: ∫u dv = uv - ∫v du

∫x3 * e2x dx

where we pick u = x3 and dv = e2x dx

du = d/dx x3= 3x2 dx

v = ∫ dv = ∫ e2x dx = e2x/2

x3 * e2x/2 - ∫ e2x/2 * 3x2 dx

= x3 * e2x/2 - 3/2 ∫ x2e2x dx

Now just need to evaluate ∫ x2e2x dx which requires another integration by parts.

We pick u = x2 and dv = e2x dx.

du = d/dx x2 = 2x dx

v = ∫ dv = ∫ e2x dx = e2x/2

x2 * e2x/2 - ∫ e2x/2 * 2x dx = x2 * e2x/2 - ∫ xe2x dx

Now have to evaluate ∫ xe2x dx which requires one final integration by parts.

Pick u = x and dv = e2x dx.

du = dx

v = ∫ dv = ∫ e2x dx = e2x/2

xe2x/2 - ∫ e2x/2 dx = xe2x/2 - 1/4 e2x

Now we have to go backwards and plug everything that needed to be plugged in with a + C at the end:

x3 * e2x/2 - 3/2 (x2 * e2x/2 - (xe2x/2 - 1/4 e2x)) + C

Feel free to reach out to me if you have questions