Browsing category integration

That pesky constant

A student asks: I've got to work out: $\int \cosec^2(x) \cot(x) \d x$. I did it letting $u = \cosec(x)$ and got an answer -- but when I did it with $u = \cot(x)$, I got something else. What gives? Ah! A substitution question! My favourite -- and it sounds

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A student asks: Why is there a $+c$ when you integrate?

A student asks: We've just started integration and I don't understand why there's always a $+c$ - I understand it's a constant, I just don't understand why it's there! Great question! The simple answer is, because constants vanish when you differentiate, they have to appear when you integrate - it's

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Integration by parts: how you do it, and why it works

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

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Why I Can See My Car works

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

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C4 Integration Quiz (tough stuff)

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

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Maths in the real world: flowerpot volumes

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

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"How do I know which method to use?" – How to Integrate

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

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A tricky C2 question…

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

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Three good reasons you divide when you integrate

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

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