evaluate the following integral. ∫6xe^2x dx

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Let $u = 6x$ and $dv=e^{2x}dx$. Then $du = 6dx$ and $v=\frac{1}{2}e^{2x}$.

Step2: Substitute into the formula

$\int 6xe^{2x}dx=6x\cdot\frac{1}{2}e^{2x}-\int\frac{1}{2}e^{2x}\cdot6dx$. Simplify to get $3xe^{2x}-3\int e^{2x}dx$.

Step3: Integrate $e^{2x}$

We know that $\int e^{2x}dx=\frac{1}{2}e^{2x}+C$. So $3xe^{2x}-3\int e^{2x}dx=3xe^{2x}-3\cdot\frac{1}{2}e^{2x}+C$.

Answer:

$3xe^{2x}-\frac{3}{2}e^{2x}+C$