Posted in core 4, integration.

Integration by parts is one of the two important integration methods to learn in C4 (the other is substitution1 ). In this article, I want to run through when you do it, how you do it, and why it works, just in case you're interested. When you use integration by

Read More →
Posted in integration.

A few weeks ago, the Mathematical Ninja revealed that he integrated trigonometric functions using a cheap mnemonic. As reader Joshua Zucker pointed out, this was most unlike the Mathematical Ninja. Had he been kidnapped? Surely not; no Ninja would ever be taken alive. Had the Mathematical Pirate infiltrated? Had someone

Read More →
Posted in core 4, integration, quizzes.

Here's a quick multiple-choice quiz about the tough stuff in C4 integration. Ready?1 Question 1: squared trig functions What method do you use to calculate $\int \sin^2(x) dx$? (Give me all four answers!) a) Parts ($u = \sin(x),~v'=\sin(x)$) b) Trig substitution ($u=\cos(2x)$) c) Split-angle formula ($\sin(A)\sin(B) = d) Parts ($u

Read More →
Posted in core 4, integration.

A reader asks: how do I figure out the volume of soil I need to fill a flowerpot? A flowerpot is a slightly peculiar shape: it's not a cone, it's not a cylinder, it's somewhere between the two. Luckily, we have a word for such shapes: it's a frustum of

Read More →
Posted in core 4, integration.

My student frowns as I write down $u = \cos(x)$. "Wait wait wait", he says, "how do you know to do a substitution?" The honest answer is, I just do. I've done integration sum after integration sum for the last 20 years and it comes naturally. However, I know that's

Read More →
Posted in core 2, graphs, integration.

I was sent this by my friend and colleague TeaKay from Blogstronomy, and I've adapted the puzzle slightly to make the sums a little bit nicer. A curve has the equation $y = (x-2)^2 - n^2$. The area bounded by the co-ordinate axes and the curve in the first quadrant

Read More →
Posted in core 4, integration.

When you integrate a function - for instance, $\cos(3x)$, you probably have to stop for a moment and think: "Do you multiply by 3 or divide when you integrate?" Some people don't even get that far, and just say "Oh, it must be $\sin(3x)$", and all of us can just

Read More →
Posted in integration, triangles, trigonometry.

There is one big-daddy among the trig identities that you need to learn right now, if you don't know it already: $$\sin^2(x) + \cos^2(x) = 1$$ This is the identity that nearly all of the others spring from. There are some more definitions: $tan(x) = \frac{\sin(x)}{\cos(x)}$, which is one of

Read More →
Posted in calculus, integration.

When you learn how to integrate by parts, you're probably told that a good rule of thumb is that if you have a power of $x$, that's going to be the thing you differentiate. That's a pretty good heuristic, but there are two places it breaks down: one, if you're

Read More →