Review Test for February 23, 2017

Hello Math Students! Here is your review test covering sections 7.1 through 7.4 to help you prepare. We have made a couple of adjustments to the original test to ensure the problems are clear and focused on the core concepts. Good luck with your studies!

Important Changes:

  • Problem #3 has been updated from section 7.1 to be more accessible and require fewer substitutions. This will allow you to focus on the key integration techniques.
  • Problem #10, originally from section 7.4, has been replaced with a problem from earlier in the section that avoids solving a system of 4 variables. This simplifies the problem and emphasizes fundamental concepts.

Review Problems

Here are some example problems similar to those you'll find on the review test. Use these with the attached PDF to help you study.

  1. Integration by Parts: Evaluate the integral $\int (x^2 + 2x) \cos(x) dx$. This problem emphasizes the application of the integration by parts formula: $\int u dv = uv - \int v du$. Remember to carefully choose your $u$ and $dv$ to simplify the integral. See the attached PDF for the full solution.
  2. Integration by Parts (Trigonometric): Solve $\int t \csc^2(t) dt$. This again uses integration by parts. Recognize that the derivative of $t$ is simple, and the integral of $\csc^2(t)$ is a standard result.
  3. Integration by Parts (Logarithmic and Exponential): Determine $\int \cos(\ln x) dx$. Here, a substitution can be helpful along with using integration by parts twice. Note how we handle the cyclic nature of the integration to arrive at the final answer.
  4. Trigonometric Integrals: Evaluate $\int \sin^5(2x) \cos^2(2x) dx$. Use trigonometric identities such as $\sin^2(x) + \cos^2(x) = 1$ to rewrite the integrand in a form that is easier to integrate. Make sure to use u-substitution as well.
  5. Definite Integrals with Trig Functions: Evaluate $\int_0^{\pi} \sin^2(x) \cos^4(x) dx$. This utilizes trigonometric identities and reduction formulas, and double angle formulas to solve.
  6. Definite Integral with tan and sec: Evaluate $\int_0^{\pi/4} \sec^6(\theta) \tan^6(\theta) d\theta$. This can be solved using a simple substitution and the pythagorean trigonometric identity.
  7. U-Substitution with Definite Integral: Evaluate $\int_0^{1/2} x \sqrt{1-4x^2} dx$. Remember to adjust the limits of integration when using u-substitution with definite integrals.
  8. Trigonometric Substitution: Evaluate $\int \frac{\sqrt{1+x^2}}{x} dx$. Here you can solve this using the substitution $x = \tan(\theta)$.
  9. Partial Fraction Decomposition: Evaluate $\int \frac{x^3 - 4x + 1}{x^2 - 3x + 2} dx$. First, perform polynomial long division. Then, decompose the resulting rational function into partial fractions.

Remember to review your notes and practice similar problems. Don't hesitate to ask questions if you're unsure about anything. You've got this!

Answer Sheet

#1-9, #10 (Note: #10 refers to the *revised* problem)