evaluate the following integral. ∫9x²(x³ + 7) dx ∫9x²(x³ + 7) dx = (type an exact answer.)

Explanation:

Step1: Use substitution method

Let $u = x^{3}+7$, then $du=3x^{2}dx$, and $9x^{2}dx = 3du$.

Step2: Rewrite the integral

The integral $\int9x^{2}(x^{3}+7)dx$ becomes $\int3u du$.

Step3: Integrate with respect to u

$\int3u du=3\times\frac{u^{2}}{2}+C=\frac{3}{2}u^{2}+C$.

Step4: Substitute back u

Substitute $u = x^{3}+7$ back, we get $\frac{3}{2}(x^{3}+7)^{2}+C$.

Answer:

$\frac{3}{2}(x^{3}+7)^{2}+C$