# C4 Integration Quiz (tough stuff)

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

Ready? ((If you’re not, you should buy my book on C4 integration.))

### 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 ($\sin(A)\sin(B) = \dots$) d) Parts ($u = \sin^2(x),~v’=1$) e) None of the above.

### Question 2: mixed trig functions

**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 ($u = \cos(x)$)

### Question 3: logarithms

**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

### Question 4: Nasty powers

**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

### Question 5: Another squared trig function

**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.

## Answers

Let’s go through them and see what our hundred people said ((Just kidding. No people were harmed in the making of this quiz.))

### Question 1: squared trig functions

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

a) Parts ($u = \sin(x),~v’=\sin(x)$) ✓ *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!*

### Question 2: mixed trig functions

**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.*

### Question 3: logarithms

**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._

### Question 4: Nasty powers

**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.*

### Question 5: Another squared trig function

**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?*