Author Archives: hangtime
Two trig identities which reduce a sin^2 or cos^2 to first-order expressions, a key skill for calculus.
Integrating Powers of Secant & Co-Secant (with Tan & Cot)
When it comes to integrating powers of trig functions, your strategy needs to include memorizing a set of rules based on whether the exponents are even or odd, and which trig functions are involved. This video covers you for even and odd exponents of secant and co-secant, often including secant & co-secant.
Integrating Powers of Tan & Cot
When it comes to integrating powers of trig functions, your strategy needs to include memorizing a set of rules based on whether the exponents are even or odd, and which trig functions are involved. This video covers you for even and odd exponents of tangent and cotangent (without secants).
Integrating Powers of Sine and Cosine
When it comes to integrating powers of trig functions, your strategy needs to include memorizing a set of rules based on whether the exponents are even or odd, and which trig functions are involved. This video covers you for even and odd exponents of sine and cosine.
Power Reducing Formulas for Sine & Cosine
With the help of these two formulas, sin^2 and cos^2 can be reduced to first-power cosines, which is something that's really helpful if: 1) you have to do really hard trig proofs, or 2) integrate a sine^2.
The power reducing formula for sine and cosine, as well as how to integrate powers of sine, cosine, tangent, co-tangent, secant, co-secant.
Trig Substitution Examples
If you haven't watched the prior two videos, you should definitely do that first, 'cause we're going to be going pretty fast here. In this video we'll just put it all together and actually solve problems from start to finish, amazing ourselves by getting correct answers.
Trig Substitution (Step 2): how to un-substitute theta for x
After taking out a perfectly good x and replacing it with a yucky trig function of Theta, at some point you're going to have to un-scramble that egg and turn things back into x's. This video is all about how to do that, and in style.
Trig Substitution (Step 1): choosing what to sub
The first step of trigonometric substitution problems is the trickiest: figuring out a trig function with which to replace a perfectly good x. In this video I show the hows and whys, and most importantly the hows, of this super tricky first step.
Also called "trig sub", a method for solving integrals with square roots in them by substituting a trig function for x.
Epsilon-Delta Limit Problems
Greek letters, arcane math, and proofs. Congratulations, math teachers, you've hit the trifecta of the least student-friendly limits section ever! Seriously, if your teacher does the section (many don't), you can bet you're in for a rough semester. In this video I'll try and take the math out of this topic, 'cause oh boy does it need it!
If your teacher covers this, you'll know it! Very painful way to start off limits.
Simplifying Complex Fractions
I get asked about these all the time because they're such scary looking problems -- fractions in the numerator and denominator of even bigger fractions -- yet one simple trick works for them every time.
Problems where you have to simplify a giant fraction which has more fractions inside the numerator and denominator.
Advanced Integration By Parts Problems
This video covers integration of arctan, and what to do when you have an exponential multiplied by a trig function. Both are tricky, but the second is more likely to show up on your quiz or test!
Integration By Parts: Natural Logs
This video is just for problems that have an ln in the integral, because it's such a common type of trick problem. Why are they tricky? Because often they're just U-sub problems in disguise, and the rest of the time they're the opposite of a normal Integration By Parts scenario. This video explains the difference, and how to defeat your teacher's trickery.
Intro to Integration By Parts
These problems are basically plug-and-chug once you get the hang of them. This first video covers most of the problems you'll see, where a power of x is multiplied by an exponential or trig function. Beware: teachers love to mess with your head by putting straightforward U-sub problems on quizzes and tests about Integration By Parts, so keep your eye out for that!
From easy to hard, this formula allows us to use u, v, du and dv to integrate the products of functions multiplied together.
Differentials
Not to be confused with "differential equations", differentials are just another type of linear approximation question they can ask. In this video I cover how to do them and how to figure out which formula to use on each type of problem, which can save you some serious work since differentials are a tiny bit easier than linear approximations.