Integration

Question 0

1
f(x) = —
x

Answer 0

∫f(x)dx = lnx

Question 1

f(x) = e(^kx)

Answer 1

1
∫f(x)dx= – e(^kx) + C
k

Question 2

f(x) = sinkx

Answer 2

1
∫f(x)dx = - — coskx
k

Question 3

f(x) = coskx

Answer 3

1
∫f(x)dx = — sinkx
k

Question 4

Integration by substitution:

y=f(u), u=g(x)

Answer 4

∫f’[g(x)]g’(x) dx = f[g(x)] + C

Question 5

Standard patterns: 
How do we integrate the form 
    f'(x)
∫k ----
    f(x)

(Something times the integration of a function divided by its original function)

Answer 5

Consider ln|f(x)| and differentiate to check how similar it is to the original equation. Adjust any constant until it is the same, then do the opposite of this to your consideration.

Question 6

Standard patterns:
How do we integrate the form:
∫ kf’(x)[f(x)]^n dx

(Something times the integration of a function times the original function)

Answer 6

Consider [f(x)]^(n+1) and differentiate to check how similar it is to the original. Adjust any constant to make the same, then do the opposite of this to your original consideration.

Question 7

Volume of revolution and area formulas:

Answer 7

A: ∫ y dx
V: pi∫ y^2 dx

Question 8

Separating the variables when dy/dx=f(x)g(y)

Answer 8

∫ 1/g(y) dy = ∫ f(x) dx