What method do you use to calculate $\int \sin^2(x) dx$? (Give me all four answers!)

A

None of the other four listed here.

Hint:

No! You mean any of the four listed here!

B

Split-angle formula

Hint:

The way you get most help from the book with

C

Trig substitution ($\cos(2x)$)

Hint:

Probably the easiest way.

D

Parts ($u = \sin(x),~v'=\sin(x)$)

Hint:

You can do! You need to do a trig identity in the next step, though.

E

Parts ($u = \sin^2(x),~v'=1$)

Hint:

Probably the most involved way - you have to do parts and a trig substitution in the second step.

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Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.

Maybe better if you show the answer for the actual maths at the end.

Just telling you but, isn’t the v=1 on the ln(x) question suppose to be v’=1? (2^x has one choice where it is v’=1)
Also, there is 80% chance that “non of the above” is not at the bottom so shouldn’t it be non of the others?

## MathbloggingAll

C4 Integration Quiz (tough stuff) http://t.co/NFQ24eRzbb

## A

Good questions thanks

Maybe better if you show the answer for the actual maths at the end.

Just telling you but, isn’t the v=1 on the ln(x) question suppose to be v’=1? (2^x has one choice where it is v’=1)

Also, there is 80% chance that “non of the above” is not at the bottom so shouldn’t it be non of the others?

## Colin

Thanks for the feedback — I’ve shied away from giving the answers to the integrals, mainly out of laziness.

Thanks especially for the corrections, I’ve fixed them 🙂

## mkami wambura

where are questions?????

## Colin

They should show up after you click the big blue ‘Start’ button.

## aayush ganesh

Differentiating log base 2 is actually fairly straightforward.

y = log2(x)

2^y = x

dx/dy = 2^y ln(2)

dy/dx = 1/((2^y)ln(2))

dy/dx = 1/xln(2)

Imo, this is easier than writing 2^x in terms of e^x and following the e^x integration by recognition rules.

The only reason it’s not used is because it’s not on the syllabus 🙁

They put differentiating and integrating any exponential but not any logarithm. Doesn’t make much sense to me.

## Colin

I’d probably do it as:

$y = \log_2(x) = \frac{\ln(x)}{\ln(2)}$

$\dydx = \frac{1}{x} \times \frac{1}{\ln(2)}$ directly.