evaluate the following integral. ∫5xe^3x dx

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Let $u = 5x$ and $dv=e^{3x}dx$. Then $du = 5dx$ and $v=\frac{1}{3}e^{3x}$.

Step2: Substitute into the formula

$\int 5xe^{3x}dx=5x\cdot\frac{1}{3}e^{3x}-\int\frac{1}{3}e^{3x}\cdot5dx$.

Step3: Evaluate the remaining integral

$\int\frac{5}{3}e^{3x}dx=\frac{5}{3}\cdot\frac{1}{3}e^{3x}+C=\frac{5}{9}e^{3x}+C$. So, $\int 5xe^{3x}dx=\frac{5}{3}xe^{3x}-\frac{5}{9}e^{3x}+C$.

Answer:

$\frac{5}{3}xe^{3x}-\frac{5}{9}e^{3x}+C$