2. (5 points) find the indefinite integral. ∫x²cos(x³ + 1) dx

Explanation:

Step1: Use substitution

Let $u = x^{3}+1$, then $du=3x^{2}dx$, and $x^{2}dx=\frac{1}{3}du$.

Step2: Rewrite the integral

The original integral $\int x^{2}\cos(x^{3}+1)dx$ becomes $\frac{1}{3}\int\cos(u)du$.

Step3: Integrate $\cos(u)$

We know that $\int\cos(u)du=\sin(u)+C$. So $\frac{1}{3}\int\cos(u)du=\frac{1}{3}\sin(u)+C$.

Step4: Substitute back

Substitute $u = x^{3}+1$ back, we get $\frac{1}{3}\sin(x^{3}+1)+C$.

Answer:

$\frac{1}{3}\sin(x^{3}+1)+C$