Calculus 2, Fall 2016

## Study tips

Don't work problems alone – find some study buddies! When multiple people study problems together, it has been scientifically demonstrated (by Uri Treisman) that

• They spend more time learning concepts

• They spend less time doing answer-checking algebra

• They more accurately guage their understanding and proficiency

• They get the help and feedback they need immediately

Don't struggle in isolation! Share your mistakes with others to speed up and improve learning!

## Useful trig identities

### Pythagorean identities

$\cos^2(\theta) + \sin^2(\theta) = 1$

$1 + \tan^2(\theta) = \sec^2(\theta)$

### Power reducing identites

$\displaystyle \cos^2(\theta) = \frac{1+\cos(2\theta)}{2}$

$\displaystyle \sin^2(\theta) = \frac{1-\cos(2\theta)}{2}$

## Useful trig identities

### Product of same types

$\cos(ax) \cos(bx) = \frac{1}{2} \bigg\lbrack \cos((a-b)x) + \cos((a+b)x) \bigg\rbrack$

$\sin(ax) \sin(bx) = \frac{1}{2} \bigg\lbrack \cos((a-b)x) - \cos((a+b)x) \bigg\rbrack$

### Product of mixed types

$\sin(ax) \cos(bx) = \frac{1}{2} \bigg\lbrack \sin((a-b)x) + \sin((a+b)x) \bigg\rbrack$

## Useful indefinite integrals

$$\displaystyle \int \sec^2(x) \, dx = \tan(x) + C$$

$$\displaystyle \int \sec(x)\tan(x) \, dx = \sec(x) + C$$

$$\displaystyle \int \sec(x) \, dx = \ln|\sec(x)+\tan(x)| + C$$

## Basic

1. $$\displaystyle \int \tan^2(3x) \, dx$$

2. $$\displaystyle \int \cos^4(t) \sin(t) \, dt$$

3. $$\displaystyle \int \tan^5(x) \sec^2(x) \, dx$$

## More involved

1. $$\displaystyle \int \sin^4(x) \, dx$$

2. $$\displaystyle \int \sin^3(4x) \cos^5(4x) \, dx$$

3. $$\displaystyle \int \tan^4(x) \, dx$$

4. $$\displaystyle \int \frac{\tan^5(x)}{\cos^3(x)} \, dx$$

5. $$\displaystyle \int \sin(2x) \cos(3x) \, dx$$

## Exploration

A nonnegative function $$p$$ with $$1$$ unit of area underneath it can be called a probability density function. An integral $$\int_a^b p(x) \, dx$$ represents the probability of some event.

1. Plot the family of curves $$p_n(x) = A_n \cos^n(x) \sin^n(x)$$, where $$A_n$$ is a constant such that the area under the curve is $$1$$ for $$0 \leq x \leq \pi/2$$.

2. Plot the family of curves $$p_n(x) = B_n \cos^n(x) \sin(x)$$, where $$B_n$$ is a constant such that the area under the curve is $$1$$ for $$0 \leq x \leq \pi/2$$.

3. Plot the family of curves $$p_n(x) = C_n \cos(x) \sin^n(x)$$ such that $$C_n$$ is a constant such that the area under the curve is $$1$$ for $$0 \leq x \leq \pi/2$$.