Chapter 8 Flashcards
sin^2(u) = ?
[1-cos(2u)]/2
cos^2(u) = ?
[1+cos(2u)]/2
If the power of sine is odd, then…
Leave one sine next to the du, and convert the remaining sines to cosines
*sin^2(u) + cos^2(u) = 1
If the power of cosine is odd, then…
Leave one cosine next to the du, and convert the remaining cosines into sines
*sin^2(u) + cos^2(u) = 1
If the powers of sine and cosine are both even, then…
Use sin^2(u) = [1-cos(2u)]/2 and cos^2(u) = [1+cos(2u)]/2 to convert the integrand to odd powers of cosine, then use the guideline for odd power cosine
If the power of secant is even, then…
Leave a secant squared next to the du, and convert the remaining secants into tangents
*sec^2(u) = 1 + tan^2(u)
If the power of tangent is odd, then…
Leave one secant and one tangent next to the du, and convert the remaining tangents to secants
*sec^2(u) = 1 + tan^2(u)
If there are no secants, and the power of tangent is even, then…
Leave a tangent squared next to the du and convert it into a secant squared factor, then expand and repeat if necessary
If there are no tangents, and the power of secant is odd, then…
Use integration by parts
√(a^2 - u^2)
- what does u equal?
- what does √(a^2 - u^2) equal?
- what does the triangle look like?
- u = asin(θ)
- √(a^2 - u^2) = acos(θ)
- opposite of θ: u, adjacent of θ: √ (a^2 - u^2), hypotenuse of θ: a
√(a^2 + u^2)
- what does u equal?
- what does √(a^2 + u^2) equal?
- what does the triangle look like?
- u = atan(θ)
- √(a^2 + u^2) = asec(θ)
- opposite of θ: u, adjacent of θ: a, hypotenuse of θ: √(a^2 + u^2)
√(u^2 - a^2)
- what does u equal?
- what does √(u^2 - a^2) equal?
- what does the triangle look like?
- u = asec(θ)
- √(u^2 - a^2) = atan(θ)
- opposite of θ: √(u^2 - a^2), adjacent of θ: a, hypotenuse of θ: u
Parts formula
u • v - {v • du
sin(2θ) = ?
2sin(θ)cos(θ)
cos(2θ) = ?
cos^2(θ) - sin^2(θ)
