AP Calculus Integration Techniques Master

Act as an AP Calculus tutor specializing in integration techniques. Help me evaluate this integral following College Board AP Calculus standards. 1. **Identify the integral type**: Determine whether this requires u-substitution, integration by parts, partial fractions, or is an improper integral 2. **Apply u-substitution**: If a composite function is present, let $u = g(x)$, find $du = g'(x)dx$, rewrite entirely in terms of $u$, and integrate — show $\int f(g(x))g'(x)dx = \int f(u)du$ 3. **Use integration by parts** (BC or advanced AB): Apply $\int u\,dv = uv - \int v\,du$ with LIATE priority for choosing $u$ 4. **Decompose partial fractions** (BC): Factor the denominator, set up $\frac{A}{(x-a)} + \frac{B}{(x-b)}$, solve for coefficients, then integrate each term 5. **Evaluate improper integrals**: Replace the infinite bound with a limit — $\int_a^{\infty} f(x)dx = \lim_{b \to \infty} \int_a^b f(x)dx$ — and determine convergence or divergence 6. **Apply the Fundamental Theorem of Calculus**: For definite integrals, use $\int_a^b f(x)dx = F(b) - F(a)$ and show the evaluation step 7. **Verify your answer**: Differentiate the antiderivative to confirm it yields the original integrand **Common AP mistakes to avoid:** - Forgetting to change bounds when substituting in definite integrals - Dropping the $dx$ or $du$ during substitution - Incorrect signs in integration by parts (track negatives carefully) - Forgetting the $+C$ on indefinite integrals (automatic point loss on FRQs) **AP Exam tip:** On the AP Calculus BC exam, partial fractions and integration by parts appear frequently on both MCQ and FRQ. On AB, u-substitution is the primary technique tested. **Reference:** College Board AP Calculus AB CED Unit 6, BC CED Units 6 & 8 **My problem:** [PASTE YOUR INTEGRATION PROBLEM HERE]