Test 1 - Covers 5.5, 6.1-6.5

*Memorize Formulas:
sin²x + cos²x = 1
tan²x + 1 = sec²x
1 + cot²x = csc²x
cos²x = (1 + cos(2x))/2
sin²x = (1 - cos(2x))/2

sin(2x) = 2sin(x)cos(x)

S 1/(u² + a²) du = (1/a) arctan(u/a) + C

For Section 5.5 you need to know:
1) "u-du" substitution.
2) All basic integrals on the course web page.
3) All basic derivatives on the course web page.

For Section 6.1 you need to know:
1) Sudv = uv - Svdu
2) It is useful to memorize:
Se^kxdx = (1/k)e^kx + C
Scos(kx)dx = (1/k)sin(kx) + C
Ssin(kx)dx = -(1/k)cos(kx) + C

For Section 6.2-Part I you need to know:
1) For Ssin^mx cosⁿx dx:
a) If sin^mx is an odd power, then let u = cos x,
du = -sin xdx
1 - u^2 = (sin x)^2

b) If cos^mx is an odd power, then let u = sin x,
du = cos xdx
1 - u^2 = (cos x)^2

c) If both sin^mx and cos^mx are even powers, use
cos²x = (1 + cos(2x))/2 sin²x = (1 - cos(2x))/2
or sometimes useful,
sinxcosx = (1/2)sin2x

2) For Stan^mx secⁿx dx:

a) If tan^mx is an odd power, then let u = secx,
du = secxtanxdx
u^2 - 1 = (tan x)^2

b) If secⁿx is an even power, then let u = tanx,
du = sec²xdx
u^2 + 1 = (sec x)^2

c) Otherwise, replace any factors of tan²x with sec²x - 1 and use the formula:
S secⁿx dx = (1/(n - 1)) sec^n-2x tan x + (n - 2)/(n - 1) S sec^n-2x dx


For Section 6.2-Part II you need to know:
1) Memorize and know to use:
Expression | Substitution
sqrt(a² - x²) | x = a sin 0
sqrt(a² + x²) | x = a tan 0
sqrt(x² - a²) | x = a sec 0

For Section 6.3 you need to know:
For f(x) = P(x)/Q(x), a rational function
1) Make sure degree of P(x) < degree of Q(x).
2) Examples:
5 = A + Cx + D
(x - 1)(x² + 5) (x - 1) (x² + 5)

5 = A + B + Cx + D
(x - 1)²(x² + 5) (x - 1) (x - 1)² (x² + 5)

5 = A + Bx + C + Dx + E
(x - 1)(x² + 5)² (x - 1) (x² + 5) (x² + 5)²

3) Multiply both sides by the denominator on the left side. Set x equal to different values to solve for A, B, C, D.

4) Rewrite the integral of f(x) as the right hand side.

For Section 6.4 you need to know:
How to use the tables in the back of the book. Here is the link for the tables in the back of the book: http://math.asu.edu/~nbrewer/CurrentCourses/MAT266/IntegralTablesStewart.pdf

For Section 6.5 you need to know:
Memorize and know to use the Left-Hand Sum, Right-Hand Sum, Midpoint Rule, Trapezoid Rule, and Simpson's Rule. Here are the rules: Rules