1. The problem is to understand and solve definite and indefinite integrals. 2. The indefinite integral of a function $f(x)$ is given by the antiderivative plus a constant $C$: $$\int f(x)\,dx = F(x) + C$$ where $F'(x) = f(x)$. 3. The definite integral from $a$ to $b$ is the net area under the curve $f(x)$ between $x=a$ and $x=b$: $$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F$ is any antiderivative of $f$. 4. Important rules: - Linearity: $$\int (af(x) + bg(x))\,dx = a\int f(x)\,dx + b\int g(x)\,dx$$ - Power rule for integrals: $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, n \neq -1$$ - Fundamental Theorem of Calculus connects definite and indefinite integrals. 5. Example: Find $$\int_0^2 (3x^2 + 2x)\,dx$$ 6. Find antiderivative: $$F(x) = \int (3x^2 + 2x)\,dx = 3\frac{x^3}{3} + 2\frac{x^2}{2} + C = x^3 + x^2 + C$$ 7. Evaluate definite integral: $$\int_0^2 (3x^2 + 2x)\,dx = F(2) - F(0) = (2^3 + 2^2) - (0 + 0) = 8 + 4 = 12$$ 8. So, the definite integral equals 12. This shows how to compute both indefinite and definite integrals step-by-step.