Test Questions on Choosing the Correct Integration Method in Calculus

📚 Quick Study Guide

• 🔍 U-Substitution: Use when you see a function and its derivative (or a constant multiple of it) within the integral. Look for something of the form $\int f(g(x))g'(x) dx$.
• 💡 Integration by Parts: Use when you have a product of two functions. Remember the formula: $\int u dv = uv - \int v du$. Choose $u$ wisely using LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to make $\int v du$ easier.
• 📝 Trigonometric Integrals: Use when you have integrals involving trigonometric functions. Employ trigonometric identities to simplify the integral. Common forms include $\int sin^m(x)cos^n(x) dx$ and $\int tan^m(x)sec^n(x) dx$.
• ➗ Partial Fraction Decomposition: Use when you have a rational function (a polynomial divided by another polynomial). Decompose the fraction into simpler fractions that are easier to integrate. This is typically used when the degree of the numerator is less than the degree of the denominator.
• ➕ Trigonometric Substitution: Use when the integral contains expressions of the form $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$. Substitute using trigonometric functions to eliminate the square root.

Practice Quiz

1. What integration method is most suitable for $\int x \cos(x) dx$?
   1. U-Substitution
   2. Integration by Parts
   3. Trigonometric Substitution
   4. Partial Fraction Decomposition
2. Which method is best for $\int \frac{2x}{x^2 + 1} dx$?
   1. Integration by Parts
   2. Partial Fraction Decomposition
   3. U-Substitution
   4. Trigonometric Integrals
3. What technique should you use for $\int \sin^3(x) \cos(x) dx$?
   1. Integration by Parts
   2. Trigonometric Substitution
   3. U-Substitution
   4. Partial Fraction Decomposition
4. Which integration method is ideal for $\int \frac{1}{x^2 - 4} dx$?
   1. U-Substitution
   2. Trigonometric Substitution
   3. Integration by Parts
   4. Partial Fraction Decomposition
5. What method would you apply to $\int \sqrt{4 - x^2} dx$?
   1. U-Substitution
   2. Integration by Parts
   3. Trigonometric Substitution
   4. Partial Fraction Decomposition
6. Which technique is most appropriate for $\int x e^x dx$?
   1. U-Substitution
   2. Integration by Parts
   3. Trigonometric Substitution
   4. Partial Fraction Decomposition
7. Which method is most suitable for $\int \frac{1}{x \ln(x)} dx$?
   1. U-Substitution
   2. Integration by Parts
   3. Trigonometric Substitution
   4. Partial Fraction Decomposition