2 - chapter 11 - further integration

Question 1

integrate e^x

Answer 1

e^x + c

Question 2

integrate 1/x

Answer 2

ln |x| + c

Question 3

integrate sec^2 x

Answer 3

tan x + c

Question 4

integrate sec x tan x

Answer 4

sec x + c

Question 5

integrate cosec x cot x

Answer 5

• cosec x + c

Question 6

integrate cosec^2 x

Answer 6

• cot x + c

Question 7

integral of f(ax + b) eg sin(2x)

Answer 7

1/a F(ax+b) + c
where F(x) if the integral of f(x)
eg -1/2cos(2x)

1/stuff differentiated* ∫outer function(inner function) + c

Question 8

integral of e^kx

Answer 8

1/k e^kx + c

Question 9

integral of 1/ax + b

Answer 9

1/a ln|ax+b| + c

Question 10

integration by substitution eg ∫sin^5x cos x

Answer 10

let u = something
sub in u
find du/dx
sub in du for dx
integrate with respect to u
the sub back in x
eg ∫sin^5x cos x
u = sin x
du/dx = cos x > dx = du *1/cos x
= ∫ u^5 du
= 1/6 u^6
= 1/6 sin^6 x + c

Question 11

integration by reverse chain rule

Answer 11

∫ f’(x) g(f(x))
f’(x)/f’(x) G(f(x)) + c

Question 12

integral of f’(x)/ f(x)

Answer 12

ln|f(x)| + c

Question 13

integration by parts

Answer 13

let one part = u
let the other = v’
find v and u’ then sub in formula
∫uv’ = uv - ∫vu’
when chosing which is u and which is v normally let u be the one u can differentiate to = 1
except if there is ln x involved make sure to to make it u ( cant integrate ln x)

Question 14

integral of ln x

Answer 14

do by parts
let u = ln x let v = x
= x ln x - x + c

Question 15

endless integration by parts

Question 16

trig identitites

Answer 16

sin^2 + cos^2 = 1
1 + tan^2 = sec^2
1 + cot^2 = cosec^2

Question 17

double angle formula sin 2x

Answer 17

2 sin x cos x

Question 18

double angle formula cos 2x

Answer 18

cos ^2 - sin^2
1 - 2 sin^2
2cos^2 - 1

Question 19

rewrite sin^2 or cos^2

Answer 19

sin^2 = (1 - cos2x)/2
cos^2 = (cos2x + 1)/2

Question 20

inetgration + partial fractions

Answer 20

when the top of a fraction is not the derivitive of the bottom - split the fraction in 2 the integrate - will be A ln() + B ln()