integration :( (1 Viewer)

[d3adgurL]

someone refresh me on how to integrate this pls:

f(x) = sin^2 x cosx

 

[wogboy]

Use this rule, it's very useful:

integral of f'(x)*f(x)^n dx = {f(x)^(n+1)}/{n+1}

note that by saying f(x)^n, I mean (f(x))^n not f(x^n)

Try to identify what f(x) and f'(x) are in this case, then apply the rule.

 

[kewpid]

let u=sinx
du/dx=cosx

therefore dx=du/cosx

sub it in

integral of cosx(sinx)^2 = integral of u^2 du
= (u^3)/3 + C
= (sinx)^3 / 3 + C

 

[Rahul]

for integration i found this in my exam, where it does not state to use a substitution, there is an easier method to find the answer, eg:
integrate the folowing,
f(x), use x= u+1
this q requires u to use substitution.

if the q is simply:
integrate,
f(x)....even though it may look like a subsitution, there is prolly another way.

i found this for 4u, would someone comment on this?
was i jus stupid not to pick up the easier method in the first place, or is this a kind of general trend?

 

[wogboy]

This is the chain rule for differentiating:

d/dx f(g(x)) = g'(x)*f'(g(x))

(simple 2U stuff)

therefore,

I g'(x)*f'(g(x)) dx = f(g(x))

Quite simple actually. You can change it around to make it easier to use:

I g'(x)*f(g(x)) dx = F(g(x))

where f(x) = F'(x) (i.e. F(x) is the primitive of f(x))

I believe this is one of the most important rules in integration. If there's a rule worth remembering, it's this one.

The question asked in this thread is proof of how useful this rule is. In this question, f(x) = x^2 and g(x) = sinx Use that rule to integrate it.