evaluate the following integral. ∫9xe^7x dx

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Let $u = 9x$ and $dv=e^{7x}dx$. Then $du = 9dx$ and $v=\frac{1}{7}e^{7x}$.

Step2: Substitute into the formula

$\int 9xe^{7x}dx=9x\cdot\frac{1}{7}e^{7x}-\int\frac{1}{7}e^{7x}\cdot9dx$.

Step3: Simplify and solve the remaining integral

$=\frac{9}{7}xe^{7x}-\frac{9}{7}\int e^{7x}dx$. Since $\int e^{7x}dx=\frac{1}{7}e^{7x}+C$, we have $\frac{9}{7}xe^{7x}-\frac{9}{7}\cdot\frac{1}{7}e^{7x}+C$.

Answer:

$\frac{9}{7}xe^{7x}-\frac{9}{49}e^{7x}+C$