Answer to Question #138060 in Calculus for Zeeshan

Question

solve the following question step bt step in detail.

∫ x²e³ˣ dx

Answer 1

Use integration by parts:

$\intop x^2e^{3x}dx=\intop x^2d\frac{e^{3x}}{3}=$

$=\frac{x^{2}e^{3x}}{3}-\intop\frac{e^{3x}}{3}dx^2=\frac{x^{2}e^{3x}}{3}-\intop\frac{2xe^{3x}}{3}dx=$

$=\frac{x^{2}e^{3x}}{3}-\frac{2}{3}\intop\ xe^{3x}dx=\frac{x^{2}e^{3x}}{3}-\frac{2}{3}\intop xd\frac{e^{3x}}{3}=$

$=\frac{x^{2}e^{3x}}{3}-\frac{2}{3}(\frac{xe^{3x}}{3}-\intop\frac{e^{3x}}{3}dx)=$

$=\frac{x^{2}e^{3x}}{3}-\frac{2xe^{3x}}{9}+\frac{2}{3}\cdot\frac{e^{3x}}{9}+C=\frac{(9x^{2}-6x+2)e^{3x}}{27}+C$