find $\\int(6x + 5)^{2}dx$.\n$\\int(6x + 5)^{2}dx=\\square$

Explanation:

Step1: Expand the integrand

$$(6x + 5)^2=(6x)^2+2\times6x\times5 + 5^2=36x^2+60x + 25$$

Step2: Integrate term - by - term

Using the power rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$:

• For $\int36x^2dx$: $36\times\frac{x^{2+1}}{2 + 1}=36\times\frac{x^3}{3}=12x^3$
• For $\int60xdx$: $60\times\frac{x^{1+1}}{1+1}=60\times\frac{x^2}{2}=30x^2$
• For $\int25dx$: $25x$

Step3: Combine the results

$\int(6x + 5)^2dx=\int(36x^2+60x + 25)dx=12x^3+30x^2+25x+C$

Answer:

$12x^3+30x^2+25x+C$