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Advanced Integration Techniques
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Advanced Integration Techniques
📚 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).
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