4. (6 points) find the indefinite integral. ∫14x(ln(x))dx

Explanation:

Step1: Use integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Let $u = \ln(x)$ and $dv=14x;dx$. Then $du=\frac{1}{x}dx$ and $v=\int14x;dx = 7x^{2}$.

Step2: Apply the formula

$\int14x\ln(x)dx=7x^{2}\ln(x)-\int7x^{2}\cdot\frac{1}{x}dx$.

Step3: Simplify the second integral

$\int7x^{2}\cdot\frac{1}{x}dx=\int7x;dx$. And $\int7x;dx=\frac{7}{2}x^{2}+C$.

Step4: Write the final result

$\int14x\ln(x)dx = 7x^{2}\ln(x)-\frac{7}{2}x^{2}+C$.

Answer:

$7x^{2}\ln(x)-\frac{7}{2}x^{2}+C$