Question

Integration by parts

1. Integral of x^3 e^4x dx
2. Integral of 2x E^2x dx

Answer 1

1.$\int x^3e^{4x}dx=x^3e^{4x}/4-(3/4)\int x^2e^{4x}dx$

$=x^3e^{4x}/4-3/4(x^2e^{4x}/4-2/4\int xe^{4x})$

$=x^3e^{4x}/4-3/4(x^2e^{4x}/4-2/4(xe^{4x}/4-e^{4x}/16)+c$

$=x^3e^{4x}/4-3x^2e^{4x}/16+(3/32)xe^{4x}-3e^{4x}/128)+c$

2.

$\int2xe^{2x}dx=2xe^{2x}/2-\int e^{2x}dx=xe^{2x}-e^{2x}/2+c$