Trig Substitutions and Important Integrarls

Question 1

substitution for a^2 - x^2

Answer 1

asin(theta)

Question 2

substitution for a^2 + x^2

Answer 2

atan(theta)

Question 3

substitution for x^2 - a^2

Answer 3

asec(theta)

Question 4

tan^2 x

Answer 4

sec^2 x - 1

Question 5

sec^2 x

Answer 5

tan^2 x +1

Question 6

integral of tan(x)

Answer 6

ln|sec x| +C

Question 7

integral of sec(x)

Answer 7

ln|sec x+ tan x|+C

Question 8

integral of tan^2(X)

Answer 8

tan x - x +C

Question 9

integral of tan^4(X)

Answer 9

(1/3) tan^3(x) -tan x +x +C

Question 10

double angle sin(2x)

Answer 10

2sinXcosX

Question 11

double angle cos(2x)

Answer 11

Cos^2 X - sin^2 X
Or
1-2sin^2 X

Question 12

d/dx 2^n

Answer 12

ln(2)*2^n

Question 13

Direct Comparison Test for Convergence

Answer 13

0<=g(x); f(x) converges if g(x) converges

Question 14

Direct Comparison Test for Divergence

Answer 14

0<=f(x),+g(x); g(x) diverges if f(x) diverges