find ∫(3x + 5)²dx. ∫(3x + 5)²dx =

Answer

Explanation:

Step1: Expand the integrand

$(3x + 5)^2=(3x + 5)(3x + 5)=9x^{2}+30x + 25$ So, $\int(3x + 5)^2dx=\int(9x^{2}+30x + 25)dx$.

Step2: Integrate term - by - term

Using the power rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$): $\int9x^{2}dx=9\times\frac{x^{2+1}}{2 + 1}=3x^{3}$, $\int30xdx=30\times\frac{x^{1+1}}{1+1}=15x^{2}$, $\int25dx=25x$.

Step3: Combine the results

$\int(9x^{2}+30x + 25)dx=3x^{3}+15x^{2}+25x+C$.

Answer:

$3x^{3}+15x^{2}+25x+C$