Written by Colin+ in core 4, integration, quizzes.

Here's a quick multiple-choice quiz about the tough stuff in C4 integration.

Ready?1

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

e) None of the above.

**What method do you use to calculate $\int \cos^9(x)\sin(x) dx$? (Give me both possibilities!)**

a) Substitution ($u = \sin(x)$)

b) Function-derivative

c) Parts ($u = \sin(x),~v' =\cos^9(x)$)

d) Parts ($u = \cos^9(x),~v=\sin(x)$)

e) Substitution ($u = \cos(x)$)

**How do you work out $\int \ln(x) dx$? (I want two methods.)**

a) Parts ($u = \ln(x),~v=1$)

b) You look it up - it's $\frac{1}{x} + C$

c) Parts ($u = 1,~v'=\ln(x)$)

d) Substitution: $x = e^u$

e) You can only do it numerically

**What method do you use to calculate $\int 2^x dx$?**

a) Parts: $u = 2^x,~v'=1$

b) Substitute $u = log_2(x)$

c) Increase the power and divide by the new power.

d) Replace $2^x$ with $e^{x\ln(2)}$.

e) You can only do it numerically

**How do you work out $\int \tan^2(x) dx$?**

a) Trig identity: $\tan^2(x) = \sec^2(x) - 1$

b) Parts: $u = \tan(x),~v'=\tan(x)$

c) Parts: $u = \tan^2(x),~v'=1$

d) Substitution: start from $\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}$

e) You can only do it numerically.

Let's go through them and see what our hundred people said2

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

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

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

b) Trig substitution ($u=\cos(2x)$)

✓ *Probably the easiest way.*

c) Split-angle formula

✓ *The way you get most help from the book with*

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

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

e) None of the above

✕ *Nope - the clue's in the question!*

**What method do you use to calculate $\int \cos^9(x)\sin(x) dx$? (Give me both possibilities!)**

a) Substitution ($u = \sin(x)$)

✕ *That's not going to work - or at least, not easily.*

b) Function-derivative

✓ *Yes, $-\sin(x)$ is the derivative of $\cos(x)$, so you can use function-derivative.*

c) Parts ($u = \sin(x),~v' =\cos^9(x)$)

✕ *I don't know how to integrate $\cos^9(x)$, and neither do you! (Do you?)*

d) Parts ($u = \cos^9(x),~v=\sin(x)$)

½ *In principle, that should work... eventually. It's a daft way to do it, though.*

e) Substitution ($u = \cos(x)$)

✓ *Yep - my preferred way to tackle these.*

**How do you work out $\int \ln(x) dx$? (I want two methods.)**

a) Parts ($u = \ln(x),~v=1$)

✓ *Yep, it drops out nicely.*

b) You look it up - it's $\frac{1}{x} + C$

✕ *No, that's differentiating*

c) Parts ($u = 1,~v'=\ln(x)$)

✕ *No, if you knew how to integrate $ln$, you wouldn't be in this mess.*

d) Substitution: $x = e^u$

✓ *Not a popular way, but a good way.*

e) You can only do it numerically

✕ *Nut-uh. You can do it numerically, of course, but it's not the only way.*

**What method do you use to calculate $\int 2^x dx$?**

a) Parts: $u = 2^x,~v'=1$

✕ *Good god, no. Have an ibuprofen.*

b) Substitute $u = log_2(x)$

½ ... sorta. Differentiating $log_2(x)$ isn't trivial, though.

c) Increase the power and divide by the new power.

✕ ✕ ✕ *YOU KILLED A KITTEN, YOU BASTARD!*

d) Replace $2^x$ with $e^{x\ln(2)}$.

✓ *Yep - messy, but it works.*

e) You can only do it numerically

✕ *That's not true.*

**How do you work out $\int \tan^2(x) dx$?**

a) Trig identity: $\tan^2(x) = \sec^2(x) - 1$:

✓ *Yep, $sec^2(x)$ integrates to $\tan(x)$. It's in the book.*

b) Parts: $u = \tan(x),~v'=\tan(x)$:

✕ *Good luck with integrating $\sec^2(x) \ln(\sec(x)\tan(x))$ in the second step.*

c) Parts: $u = \tan^2(x),~v'=1$

✕ *This ends up as $x\tan^2(x) - \int 2x \tan^2(x)sec(x) dx$. I reckon it's possible, but I don't fancy it.*

d) Substitution: start from $\tan(2x) = \frac{2\tan(x)}{1-\tan^2(x)}$

✕ *It's a nice thought, but no. You end up with a $\frac{\tan(2x)}{\tan(x)}$ to integrate, which is no good at all.*

e) You can only do it numerically.

✕ *Who gave you that idea?*

- If you're not, you should buy my book on C4 integration. [↩]
- Just kidding. No people were harmed in the making of this quiz. [↩]