evaluate the following integral. ∫ 11xe^(-7x) dx

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Let $u = 11x$ and $dv=e^{-7x}dx$. Then $du = 11dx$ and $v=-\frac{1}{7}e^{-7x}$.

Step2: Substitute into the formula

$\int 11xe^{-7x}dx=11x\left(-\frac{1}{7}e^{-7x}\right)-\int\left(-\frac{1}{7}e^{-7x}\right)\times11dx$.

Step3: Simplify the first term and evaluate the second integral

The first term is $-\frac{11}{7}xe^{-7x}$. The second integral: $\int\left(-\frac{11}{7}e^{-7x}\right)dx=\frac{11}{49}e^{-7x}+C$.

Answer:

$-\frac{11}{7}xe^{-7x}+\frac{11}{49}e^{-7x}+C$