U-Substitution Definite Integral Examples with Detailed Solutions

📚 Quick Study Guide

• 🔑 U-Substitution: A technique to simplify integrals by substituting a function $u = g(x)$ and its derivative $du = g'(x)dx$.
• 🔄 Definite Integrals: Integrals with upper and lower limits of integration, resulting in a numerical value.
• 🪜 Steps for U-Substitution with Definite Integrals:
  1. 🧮 Choose a suitable $u = g(x)$.
  2. ➗ Find $du = g'(x)dx$.
  3. ✍️ Rewrite the integral in terms of $u$.
  4. 🎯 Change the limits of integration: If $x = a$, then $u = g(a)$, and if $x = b$, then $u = g(b)$.
  5. ∫ Evaluate the new integral with respect to $u$.
  6. 📈 The result is a numerical value.
• 📝 Formula: $\int_{a}^{b} f(g(x))g'(x) dx = \int_{g(a)}^{g(b)} f(u) du$

Practice Quiz

1. What is the first step when evaluating a definite integral using u-substitution?
   1. Choose a suitable $u = g(x)$.
   2. Evaluate the integral directly.
   3. Find the derivative of the integrand.
   4. Ignore the limits of integration.
2. If $u = x^2 + 1$, what is $du$?
   1. $du = x dx$
   2. $du = 2x dx$
   3. $du = dx$
   4. $du = 2 dx$
3. When using u-substitution with a definite integral, what must be done with the limits of integration?
   1. The limits stay the same.
   2. The limits are ignored.
   3. The limits must be converted to $u$ values.
   4. The limits are squared.
4. Evaluate $\int_{0}^{1} 2x(x^2 + 1)^2 dx$ using u-substitution. Let $u = x^2 + 1$.
   1. $\frac{7}{3}$
   2. $\frac{1}{3}$
   3. $\frac{8}{3}$
   4. $\frac{2}{3}$
5. Evaluate $\int_{0}^{\pi/2} \sin(x) \cos(x) dx$ using u-substitution. Let $u = \sin(x)$.
   1. 0
   2. $\frac{1}{2}$
   3. 1
   4. 2
6. Evaluate $\int_{1}^{e} \frac{\ln(x)}{x} dx$ using u-substitution. Let $u = \ln(x)$.
   1. $\frac{1}{2}$
   2. 1
   3. 0
   4. 2
7. What is the correct substitution for evaluating $\int_{0}^{2} x\sqrt{4 - x^2} dx$?
   1. $u = \sqrt{4-x^2}$
   2. $u = 4 - x^2$
   3. $u = x^2$
   4. $u = x$