7 Integration Flashcards
integrate x^n
x^(n+1) /n+1 +c
WHAT DON’T YOU FORGET
+C
indefinite integral of e^x
e^x +c
indefinite integral of cosx
sinx +c
find the area of 5+ 2sin3x betweenx=0 and x=1/6pi
5/6pi +2/3
when is there no indefinite integral for 1/x
when x is a negative
why is y=ln(-x) possible
-x is positive when x values are negative
y= ln(-x), dy/dx=
-(1/-x) = 1/x
if x>0 and x<0 then
x (=/) 0
5–>2 integrate for 3-2/x
3x-2ln|x|
9-2ln|x|
3—>1 integrate (x^2 +1)/x
x^2/x +1/x
1/2x^2 +ln|x|
4+ln3
integrate sqrrt(1-x^2)
u= 1-x^2
du/dx = -2x
du=-sxdx
S x(u^1/2)(-2x)dx -1/2S(1-x^2)^1/2 dx
- 1/2 x2/3(1-x^2)^3/2 +c
- 1/3(1-x^2)^3/2 +c
when you notice that the numerator is the derivative of the denominator
f’(x)/f(x)
du/dx= f’(x)
du= f’(x) dx
=ln|f(x)| +c
S x^3/(X^4+1)dx
make so that it is f’(x)/f(x) —–> 1/4 S 4x^3/x^4+1 dx
1/4ln|x^4 +1| +c
integration by parts
integrating backwards
uv+c = udv/dx + vdu/dx
uv+c =Sudv/dx + Svdu/dx
udv/dx = uv- S vdu/dx dx +c
S xcos3x dx
u=x dv/dx =cos3x
du/dx= 1 v=1/3sin3x
x/3sin3x + S(1/3sin3x)(1)dx +c
x/3sin3x +1/9cos3x +c
2—>0 S xe^3x dx
u= x dv/dx= e^3x
du/dx=1 v=1/3e^3x
x/3 e^3x - S1/3e^3x dx
2/3e^6 -(1/9e^6 -1/9e^0)
two standard integrals
S 1/1+x^2
x=tanu
dx/du= sec^2u
dx=sec^2udu
1/sec^2u xsec^2udu =sec^2u/sec^2u du
S1du =u +c
x= tanu u= tan^-1x
S 1/(1+x^2) dx = tan^-1x +c
u= 5x to find S 1/sqqrt(1-25x^2) dx
S sqqrt(1-u)/sqqrt(1-u) du
S1du= u+c = 5x +c
S 1/ (a^2 +x^2) dx =
1/a tan^-1 (x/a) +c
S1/(sqrrt(a^2 -x^2) dx = sin^-1 (x/a) +c
S 1/(9-x^2) dx
x = 3tanu
dx/du =3sec^2u
dx = 3sec^2u du
S 3sec^2u/9sec2u du
=S1/3du =1/3u +c
u= tan^-1(x/3)
1/3tan^-1(x/3) +c
S 1/(Ax+b)
1/a ln|ax+b| +c
S x/4+x^2 dx
1/2 S2x/(4+x^2) dx
1/2ln|4+x^2| +c
S x/sqrrt(4-x^2) dx
u=4-x^2
du/dx =-2x
dx= -1/2x du
Sxx/sqrrt(4-x^2) x 1/2x du
S 1/2sqrrt(4-x^2) du
- 1/2 S u^-1/2 du
- 1/2 x 2u^1/2 = -sqrrt (4-x^2) +c
