Integration Jeopardy

∫1/x^n dx

1

∫1/x dx

ln(|x|)+c

2

∫1/x^2 dx

-1/x + C

3

∫1/x^3 dx

-1/(2x^2)+C

4

∫1/x^4 dx

-1/(3x^3)+C

5

∫1/x^5 dx

-1/(4x^4)+C

∫1/(x^n+1) dx(be scared)

1

∫1/(x+1) dx

ln(|x+1|)+C

2

∫1/(x^2+1) dx

arctan(x)+C

3

∫1/(x^3+1) dx

(-ln(x^2-x+1)-2(ln(|x+1|)+sqrt(3)arctan((2x-1)/sqrt(3)))/6+C

4

∫1/(x^4+1) dx

(ln(x(x+sqrt(2))+1)-ln(x(x-sqrt(2))+1)+2(arctan(sqrt(2)x+1)+arctan(sqrt(2)x-1)))/(2^(5/2))+C

5

∫1/(x^5+1) dx

(4ln(|x+1|)+(sqrt(5)-1)ln(x(2x+sqrt(5)-1)+2)+(-sqrt(5)-1)ln(x(2x-sqrt(5)-1)+2)+2^(3/2)*sqrt(sqrt(5)+5)*arctan((4x+sqrt(5)-1)/(sqrt(2)*sqrt(sqrt(5)+5))/20 - (sqrt(5)-5)arctan((4x-sqrt(5)-1)/(sqrt(2)sqrt(5-sqrt(5)))/(5sqrt(2)sqrt(5-sqrt(5)))+C

Integration by Parts

1

∫x/x dx

(given ∫ln(x) = x(ln(x)-1)+C)

x + C

2

∫u dv + ∫v du

uv or uv+C

3

∫x*sin(x) dx

sin(x)-x cos(x) + C

4

∫x*ln(x) dx

x^2*(2 ln(x)-1)/4 +C

5

∫cos^2(x)-sin^2(x)

cos(x)sin(x)+C

Definite Integrals

1

x from 0 to inf

∫e^(-100x) dx

1/100

2

x from -5 to 5

∫ (cos(x))^10 * x^7 dx

0

3

x from 0 to 0

∫ 2 sec^2(x) tan(x) dx

0

4

x from 0 to 2

∫ x^7 dx

32

5

x from -inf to inf

∫e^(-x^2) dx

sqrt(pi)

Trig Identities

1

∫cos(x) dx

sin(x) + C

2

∫sin(x) dx

-cos(x) + C

3

∫sin^2(x) + cos^2(x) dx

1

4

∫cos^2(x) dx

(cos(x)sin(x)+x)/2+C

5

∫2 sec^2(x) tan(x) dx

tan^2(x)+C