Integral Calc Integration & Volumes

Find the derivatives for the following:

1) f(x) = sin (x)

2) f(x) = cos (x)

1. cos (x)

2. -sin(x)

Find the derivatives for the following:

1) f(x) = tan (x)

2) f(x) = cot (x)

1. sec² (x)

2. -csc² (x)

Find the derivatives for the following:

1) f(x) = cos (x)

2) f(x) = csc (x)

3) f(x) = cot (x)

1. -sin (x)

2. -csc(x) cot (x)

3. -csc² (x)

Find the derivatives for the following:

1) f(x) = sec (x)

2) f(x) = csc (x)

1. sec (x) tan (x)

2. - csc (x) cot (x)

Find the derivatives for the following:

1) f(x) = log_a(x)

2) f(x) = ln (x)

1. 1/[x*ln (a)]

2. 1/x

Find the derivatives for the following:

1) f(x) = arcsin(x)

2) f(x) = arcos(x)

3) f(x) = arctan(x)

1. 1/(√1-x²)

2. -1/(√1-x²)

3. 1/1+x²

a(t) = 

v’(t) = s’’(t)

What is the antiderivative of the following?

xⁿ

1/(n+1) * xⁿ⁺¹

Find the antiderivatives for the following:

1) f(x) = sin (kx)

2) f(x) = cos (kx)

-1/(k) cos (kx)

1/(k) sin (kx)

What is the antiderivative of the following?

1/x

ln |x|

What is the antiderivative of the following?

e^kx

1/(k) e^kx

What is the antiderivative of the following?

1/√(1-k²x²)

1/k arcsin (kx)

What is the antiderivative of the following?

a^kx

1/(k ln (a)) * a^kx

What is the antiderivative of the following?

sec²(kx)

1/k tan (kx)

What is the integral of the following?

tan (u) du

ln |sec (u)| +C

What is the integral of the following?

sec (u) du

ln |sec (u) + tan (u)| +C

What is the integral of the following?

cot (u) du

ln |sin (u)| +C

What is the integral of the following?

csc (u)du

-ln |csc (u) + cot (u)| + C

Finish these Pythagorean Trig Identities

1. sin²(x) + cos²(x) =

2. sec²(x) =

3. csc²(x) =

1. = 1

2. = tan²(x) + 1

3. = cot²(x) + 1

Finish these Power Reduction Formulas

1. sin²(x) =

2. cos²(x) =

1. = ½ (1 - cos (2x))

2. = ½ (1+cos(2x))

Finish the Double Angle formula

sin (2x) =

= 2 sin(x) cos(x)

What trig substitution do you use for each form:

• a² - x²

• a² + x²

• x² - a²

• x = a sin θ

• x = a tan θ

• x = a sec θ

In ∫sin^m(x) cosⁿ(x) dx

How do you proceed if m & n are both even? What if m is odd? What if n is odd?

• if both even, use power reduction

• if m is odd, break off a sin and use sin²(x) = 1 - cos²(x)

• if n is odd, break off a cos and use cos²(x) = 1 - sin²(x)

cot (Θ)

adj/opp

OR

cos (Θ)/sin (Θ)

csc (Θ)

hyp/opp

OR

1/sin(x)

sec (Θ)

hyp/adj

OR

1/cos(Θ)

Volumes - Disk Method

Volumes - Washer Method

Volumes - Shell Method

How do you integrate something that has functions as bounds?

Use extension of 2nd FTC

Average value RoC formula

integral/interval OR

area

———

b - a

Summation Formulas

Riemann Sum Form

Define all the variables in Riemann Sum formula.

Δx = (b-a)/n

xi* = any number in the interval, usually will be told to use upper or lower

= Xo + iΔX (for righthand endpoints, bc [Xi-1, Xi] )

True or False. If taking the integral of a function f(x), from -a to a, the answer will be 0.

False, not always.

If odd function (like sin), it equals 0.

If even function (like cos), it equal 2*the integral from 0 to a.

Mean Value Theorum for Integration

IF func. is continuous on interval [a,b] THEN you can find a number c (the x value) that gives the avg value (the y-value; integral / interval)

Order of integration techniques from easiest to hardest

U-sub

IBP (ILATE)

Trig IDs

Trig sub

Partial Fractions

Whats the process for finding the area between curves?

1. draw graph, if possible

2. Find intersection(s)

3. If multiple, break up the integral and bounds

4. Integrate the difference between the bottom function and the top function for each integral term

(bottom and top func might switch - its ok)

True or False. If you’re using IBP on a composite function, set u to be the inside function.

False. You can’t separate a composite function. Set u to be the whole composite func, and set dv=dx