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

• They get better grades

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

Pythagorean identities

\[\cos^2(\theta) + \sin^2(\theta) = 1\]

\[1 + \tan^2(\theta) = \sec^2(\theta)\]

Power reducing identites

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

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

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

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

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

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

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

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

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

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

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

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 \(0 \leq x \leq \pi/2\).