Test Questions for U-Substitution in Definite Integrals (Calculus)

📚 Quick Study Guide

🔍 When dealing with definite integrals and u-substitution, remember to change the limits of integration to be in terms of $u$. If your original limits are $a$ and $b$ (for $x$), and you let $u = g(x)$, then your new limits will be $g(a)$ and $g(b)$.

💡 The formula for u-substitution in definite integrals is: $\int_{a}^{b} f(g(x))g'(x) dx = \int_{g(a)}^{g(b)} f(u) du$

📝 Key steps:
• Choose a suitable $u$ (often the inner function).
• Find $du$.
• Change the limits of integration.
• Evaluate the new integral with respect to $u$.

Practice Quiz

Question 1: $\int_{0}^{2} x(x^2 + 1)^3 dx = ?$ where $u = x^2 + 1$ A) 60 B) 50 C) 40 D) 30 Question 2: $\int_{0}^{1} x e^{x^2} dx = ?$ where $u = x^2$ A) $\frac{1}{2}(e - 1)$ B) $e - 1$ C) $\frac{1}{2}e$ D) $e$ Question 3: $\int_{1}^{2} \frac{2x}{x^2 + 1} dx = ?$ where $u = x^2 + 1$ A) $\ln(5/2)$ B) $\ln(5/3)$ C) $\ln(2)$ D) $\ln(3)$ Question 4: $\int_{0}^{\sqrt{\pi}} x \cos(x^2) dx = ?$ where $u = x^2$ A) 0 B) 1 C) 2 D) -1 Question 5: $\int_{0}^{\pi/2} \sin(x) \cos(x) dx = ?$ where $u = \sin(x)$ A) 1/2 B) 1 C) -1/2 D) -1 Question 6: $\int_{0}^{1} \frac{x}{x^2+4} dx = ?$ where $u = x^2+4$ A) $\frac{1}{2} \ln(\frac{5}{4})$ B) $\ln(\frac{5}{4})$ C) $\frac{1}{4} \ln(\frac{5}{4})$ D) $\frac{1}{3} \ln(\frac{5}{4})$ Question 7: $\int_{0}^{\pi/4} \tan(x) \sec^2(x) dx = ?$ where $u = \tan(x)$ A) 1/2 B) 1 C) -1/2 D) -1

Click to see Answers

1. B) 50
2. A) $\frac{1}{2}(e - 1)$
3. A) $\ln(5/2)$
4. A) 0
5. A) 1/2
6. A) $\frac{1}{2} \ln(\frac{5}{4})$
7. A) 1/2