Chapter 7 Test Review

Hello everyone! As you prepare for the upcoming Chapter 7 test, I want to provide you with the resources you need to succeed. This chapter has focused on various integration techniques. Here's a breakdown of what you can expect and some practice problems to help you solidify your understanding.

Key Concepts to Review

  • U-Substitution: This is your go-to method when you can identify a function and its derivative (or a constant multiple thereof) within the integrand.
  • Integration by Parts: Use this technique when you have a product of two functions, and u-substitution doesn't work. Remember the formula: $\int u dv = uv - \int v du$. LIATE (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) is a helpful mnemonic for choosing $u$.
  • Trigonometric Integrals: Master the integrals involving powers of sine, cosine, tangent, and secant. Use trigonometric identities to simplify the integrals. For example, if you have $\int sin^m(x) cos^n(x) dx$, consider these cases:
    • If $n$ is odd, save one cosine and use the identity $cos^2(x) = 1 - sin^2(x)$.
    • If $m$ is odd, save one sine and use the identity $sin^2(x) = 1 - cos^2(x)$.
    • If both $m$ and $n$ are even, use the half-angle identities: $sin^2(x) = \frac{1 - cos(2x)}{2}$ and $cos^2(x) = \frac{1 + cos(2x)}{2}$.
  • Trigonometric Substitution: When your integrand involves expressions like $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$, try trigonometric substitution.
    • For $\sqrt{a^2 - x^2}$, let $x = a sin(\theta)$.
    • For $\sqrt{a^2 + x^2}$, let $x = a tan(\theta)$.
    • For $\sqrt{x^2 - a^2}$, let $x = a sec(\theta)$.
  • Partial Fraction Decomposition: Use this for rational functions (polynomial divided by polynomial). Decompose the rational function into simpler fractions that you can integrate more easily.

Practice Problems

Here are a few practice problems similar to what you might see on the test. Solutions are provided so you can check your work.

  1. $\int \frac{x^2}{x^3 + 1} dx$
  2. $\int sin^7(x) cos^5(x) dx$
  3. $\int ln(\sqrt{x}) dx$
  4. $\int \frac{x^2 - x + 6}{x^3 + 3x} dx$
  5. $\int tan^2(x) sec^4(x) dx$
  6. $\int \frac{x}{\sqrt{36 - x^2}} dx$
  7. $\int (x^2 + 2x) cos(x) dx$
  8. $\int \frac{2}{2x^2 + 3x + 1} dx$
  9. $\int \frac{\sqrt{x^2 - 1}}{x^4} dx$
  10. $\int sin(x) \sqrt{1 + cos(x)} dx$

Good luck with your studying, and I'll see you in class! Remember to review your notes, practice consistently, and don't hesitate to ask questions.