Exam 2 Flashcards
∫x^n dx
1/n+1 X^n+1 +C
∫1/x dx
ln|x|+C
∫1/(ax+b)dx
1/a ln|a+b|+C
∫lnax dx
x(lnax -1)
∫a^x dx
a^x /lna +C
∫e^x dx
e^x +c
∫cscxdx
ln|tanx/2|+C
∫csc^2(ax)dx
-1/a cot(ax)+C
∫sin(ax) dx
-1/a cos(ax)+C
∫secx *cscx dx
ln|tanx|+C
∫cos(ax)
1/asin(ax)+C
∫tan(ax) dx
-1/a ln(cos(ax)) +C
∫sec^2 (ax) dx
1/a tan(ax) +C
∫sec(x)tan(x) dx
secx +C
∫secx dx
ln|secx+tanx|+C
work
(a to b)∫ F(x) dx
Force of spring
spring constant * distance stretched
Force of y
mass *gravity
what is density * volume
mass
∫lnx dx
x(lnx-1)+C
∫du / sqrt(a^2 - u^2)
sin^-1 (u/a)+C
∫log(base a)x
x/lna (lnx-1)+C
∫du/ a^2 +u^2
1/a tan^-1 (u/a)+C
work for water
(a to b) ∫densitygravityarea*x
∫du/ u(sqrt(u^2 -a^2)
1/a sec^-1(u/a) +C
∫udv
uv -∫ vdu +C
LIATE
Log , inverse trig, alegriaic, trig, exponential
integrating sin+cos products with odd power trig
- write odd as even power * 1 of it
2.get it in terms of the other - integrate making u the higher power
integrating sin+cos products with both even powers
- double angle identities to get into polynomial of cos(2x)
- apply strategies to things with powers greater than 1
double angle cos^2(x)
1/2 (1+cos(2x)
double angle for sin^2(x)
1/2 (1-cos(2x))
identity for cos^2(x)
1-sin^2(x)
identity for sin^2(x)
1-cos^2(x)
sin(ax)cos(bx)
1/2 sin((a+b)x +1/2 sin ((a-b)x)
cos(ax)cos(bx)
1/2 cos((a+b)x +1/2 cos ((a-b)x)
tan is even sec is odd
- write tan in terms of secx
- write polynomial in sec
3 reduction method
integrating tan+sec products with sec being even
1.split out a sec^2(x) while keeping secx to even power
2.u sub with tan as u
integrating sin+cos products with odd power tan
- split out secxtanx
2 write tan in even power - u is secx
identity of sec^2(x)
tan^2(x) +1
tan reduction formula
tan^n (x) dx
1/n-1 tan^(n-1) (x) - ∫tan^ (n-2) (x)dx
sec reduction formula
sec^n (x) dx
1/n-1 sec^(n-2) (x)tan(x) + n-2/(n-1)
∫tan^ (n-2) (x)dx
trig sub a^2-x^2
x = asin(theta)
trig sub a^2 + x^2
x = atan(theta)
trig sub x^2 - a^2
x = a(sec(theta)
