Integration reduction formula (1 Viewer)

[cutemouse]

Why is the integral of tan^n-2x.sec²xdx = tan^n-1x/(n-1) and the integral of cot^n-2x.cosec²xdx = cot^n-1/(n-1) ?

How am I supposed to work this out?

Thanks

 

[alakazimmy]

tan^n-2x.sec²xdx = tan^n-1x/(n-1)

The derivative of tan x is sec²x, so if you use substitution of u= tan x, u get du = sec²xdx
Then you are just integrating a power function of u^n-2
Hence, you get that result.

It's the identical result for the other one except you replace tan x with cot x and sec x with cosec x

 

[Trebla]

Reverse chain rule:
sec²x is the derivative of tan x
Hence ∫ (tan x)^{n - 2}sec²x dx = (1/(n - 1)∫ (n - 1)(tan x)^{n - 2}sec²x dx
= (tan x)^{n - 1} / (n - 1) + c

If you can't see this, use the substitution u = tan x.

Similar case applies with the cot x and cosec x, except that expression you typed up is missing a negative sign.

∫ (cot x)^{n - 2}cosec²x dx = - 1/(n - 1)∫ (n - 1)(cot x)^{n - 2}(- cosec²x) dx
= - (cot x)^{n - 1} / (n - 1) + c

Again, if you can't see this, use the substitution u = cot x.

 

[jet]

You need to know in 4 unit all of your trig derivatives. Your teacher should have gone over these with you. Once you know them, then the use of the reverse chain rule naturally follows.
Derivatives of Trigonometric Functions

 

[cutemouse]

Hmm okay thanks.. What is the basic rule for the reverse chain rule?

Thanks

 

[Timothy.Siu]

Hmm okay thanks.. What is the basic rule for the reverse chain rule?

Thanks

just think backwards i guess.

e.g. integral of sin⁵ x cos x. when u see that u immediately see that cos x is the derivative of sin x so u can just integrate it without substitutions.

=sin⁵x d(sin x)

 

[alakazimmy]

Hmm okay thanks.. What is the basic rule for the reverse chain rule?

Thanks

The reverse chain rule is integration by substitution, without actually doing the substitution, if that made any sense... lol

 

[jet]

It's just the chain rule, but in reverse. So if you think, d/dx(f(g(x))) = f'(g(x)) • g'(x), then, if you have an integral, ∫f'(g(x)) • g'(x) dx then it equals f(g(x)).

 

[Drongoski]

It's just the chain rule, but in reverse. So if you think, d/dx(f(g(x))) = f'(g(x)) • g'(x), then, if you have an integral, ∫f'(g(x)) • g'(x) dx then it equals f(g(x)).

How do you get the baby integral sign and the bold dot ?? Where can i find the other special symbols? Thank you.

 

[cutemouse]

Thanks alot jetblack! +1

How do you get the baby integral sign and the bold dot ?? Where can i find the other special symbols? Thank you.

Just copy it, or use the alt commands.

 

[jet]

They're just shortcuts on my mac keyboard. I hold the option key and each letter is also a symbol. it just so happens that option-b is the ∫ symbol whilst option-8 is the • symbol. In windows go to the character palette in accessories.

 

[Drongoski]

They're just shortcuts on my mac keyboard. I hold the option key and each letter is also a symbol. it just so happens that option-b is the ∫ symbol whilst option-8 is the • symbol. In windows go to the character palette in accessories.

Thank you for yr info; jm01 too.