- #1
gb7nash said:10a looks fine.
For 10b, take advantage of the following trig identity:
cosh2 x = [1 + cosh (2x)]/2
gb7nash said:http://planetmath.org/encyclopedia/HyperbolicIdentities.html
As far as knowing which ones to remember, it would depend on what you're trying to do and what class you're in. For a lot of integral problems, the identities I usually see are sin2 + cos2 = 1 (and any other pythagorean identities), and half/double angle formulas.
The process for solving difficult integrals involves breaking down the integral into smaller, simpler parts, using substitution or integration techniques, and applying basic algebraic manipulations. It may also involve using special integration techniques such as integration by parts or trigonometric substitution.
Some common techniques for solving difficult integrals include substitution, integration by parts, trigonometric substitution, partial fractions, and using tables of integrals. It is important to select the appropriate technique based on the form of the integral.
The best way to determine which technique to use for a difficult integral is to first simplify the integral as much as possible by using basic algebraic manipulations. Then, look for patterns and similarities to known integrals and choose the appropriate technique based on these patterns.
Common mistakes when solving difficult integrals include forgetting to substitute for variables, applying the wrong integration technique, forgetting to include the constant of integration, and making algebraic errors. It is important to carefully check each step and double check the final answer for accuracy.
Yes, technology can be used to solve difficult integrals. Many graphing calculators and computer software have built-in integration capabilities that can solve a wide range of integrals. However, it is still important to understand the process and techniques for solving integrals by hand, as technology may not always be available.