Question

Integrate the following with respect to x.

x³e^3x

Answer 1

= ${\displaystyle \int x^3{\text{e}}^{3x} \text{d}x}$

= ${\displaystyle \int \text{udv}}$ = uv – u¹v₁ + u¹¹v₂ – u¹¹¹v₃ ………

${\displaystyle \int x^3{\text{e}}^{3x} \text{d}x=\left(x^3\right)\left(\frac{{\text{e}}^{3x}}{3}\right)-\left(3x^2\right)\left(\frac{{\text{e}}^{3x}}{9}\right)+6x\left(\frac{{\text{e}}^{3x}}{27}\right)-+\left(\frac{{\text{e}}^{3x}}{81}\right)+\text{c}}$

=  ${\displaystyle {\text{e}}^{3x}[\frac{x^3}{3}-\frac{3x^2}{9}+\frac{6x}{27}-\frac{6}{81}]+\text{c}}$

= ${\displaystyle {\text{e}}^{3x}[\frac{x^3}{3}-\frac{x^2}{3}+\frac{2x}{9}-\frac{2}{27}]+\text{c}}$

Successive derivatives	Repeated Integrals
Take  u = x3 uI = 3x2 uII = 6x uIII = 6 uIV = 0	and dv = e3x  ${\displaystyle \int \text{dv}=\int {\text{e}}^{3x} \text{d}x}$ v = ${\displaystyle \frac{{\text{e}}^{3x}}{3}}$ v1 = ${\displaystyle \frac{{\text{e}}^{3x}}{9}}$ v2 = ${\displaystyle \frac{{\text{e}}^{3x}}{27}}$ v3 = ${\displaystyle \frac{{\text{e}}^{3x}}{81}}$