Advanced Integration Techniques

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📚 Part 1: Multiple Choice (choose the best answer)

1. Evaluate: ∫ x e^x dx

e^x(x + 1) + C

e^x(x - 1) + C

x e^x + C

e^x + C

2. Find an antiderivative: ∫ ln x dx (x > 0)

x ln x + C

x ln x - x + C

(ln x)^2/2 + C

ln(x)/x + C

3. For ∫ (5x + 7)/(x^2 + 3x + 2) dx, the best immediate step is:

Factor the denominator and use partial fractions

Substitute u = x^2 + 3x + 2

Complete the square on the denominator

Perform polynomial long division first

4. Which method is most appropriate for ∫ e^{2x} sin 3x dx?

Single substitution u = 2x

Integration by parts applied twice (or complex exponential method)

Partial fractions

Trigonometric substitution

✏️ Part 2: Short Answers — Work clearly (show method)

5. Compute: ∫ x^2 e^x dx. Indicate your choice of u and dv, and give the final antiderivative.

6. Evaluate: ∫ e^{2x} sin 3x dx. Use integration by parts (or complex exponential) and present the simplified result.

7. Integrate by partial fractions: ∫ (5x + 7)/(x^2 + 3x + 2) dx. Factor the denominator first.

8. Find ∫ (3x^2 + 2x + 1)/(x + 1)^3 dx. Decompose appropriately and give the antiderivative.

9. Evaluate: ∫ (7x + 5)/(x^2 + 4x + 5) dx. Complete the square and split into standard forms.

10. Challenging: ∫ x^3 ln(x^2 + 1) dx. Suggest a substitution or parts strategy, then compute the integral (show key steps).

Advanced Integration Techniques

Advanced Integration Techniques

📚 Part 1: Multiple Choice (choose the best answer)

✏️ Part 2: Short Answers — Work clearly (show method)

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