$\\int(12x + 14)(3x^{2}+7x - 1)^{5}dx=$

Explanation:

Step1: Use substitution method

Let $u = 3x^{2}+7x - 1$, then $du=(6x + 7)\times2dx=(12x + 14)dx$.

Step2: Rewrite the integral

The integral $\int(12x + 14)(3x^{2}+7x - 1)^{5}dx$ becomes $\int u^{5}du$.

Step3: Integrate $u^{5}$

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), we have $\int u^{5}du=\frac{u^{6}}{6}+C$.

Step4: Substitute back $u$

Substitute $u = 3x^{2}+7x - 1$ back into the result, we get $\frac{(3x^{2}+7x - 1)^{6}}{6}+C$.

Answer:

$\frac{(3x^{2}+7x - 1)^{6}}{6}$