evaluate the following integral. ∫(4x + 5)ln(8x)dx

Explanation:

Step1: Use integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Let $u = \ln(8x)$ and $dv=(4x + 5)dx$. Then $du=\frac{1}{x}dx$ and $v=\int(4x + 5)dx=2x^{2}+5x$.

Step2: Apply the integration - by - parts formula

$\int(4x + 5)\ln(8x)dx=(2x^{2}+5x)\ln(8x)-\int(2x^{2}+5x)\frac{1}{x}dx$.

Step3: Simplify the second integral

$\int(2x^{2}+5x)\frac{1}{x}dx=\int(2x + 5)dx$.

Step4: Evaluate the integral $\int(2x + 5)dx$

$\int(2x + 5)dx=x^{2}+5x+C$.

Step5: Write the final result

$\int(4x + 5)\ln(8x)dx=(2x^{2}+5x)\ln(8x)-(x^{2}+5x)+C$.

Answer:

$(2x^{2}+5x)\ln(8x)-x^{2}-5x + C$