question on recurring integrals (1 Viewer)

[FD3S-R]

how do i do a question like this

If In = integral (tan^(n)x dx for n greater than/equal to 0, show that In = 1/(n-1)tan^(n-1)x-I(n-2) for n greater than/equal to 2.

Exercise 5.5 q15 i think its cambridge

 

[Captain pi]

Break tanⁿ(x) into tan^{n – 2}(x) and tan²x.

Use the fact that tan²(x) + 1 = sec²(x).

Recognize that tan^{n – 2}(x).sec²(x) can be operated on by the reverse chain rule.

Have fun!

Love,

pi.

 

[Ogden_Nash]

int{tanⁿx dx} = int{tan^n-2x * tan²x dx}
= int{tan^n-2x * (sec²x - 1) dx}
= int{tan^n-2x * sec²x dx} - int{tan^n-2x dx}
= tan^n-1x/n-1 - I_n-2 .... QED

This question is a bit tricky because you don't have to use integration by parts.

[Edit] Nice work Pi, you're too quick.

 

[FD3S-R]

bah thanks alot guys, i was trying to use tanx and tan^(n-1)x just got introduced to these questions had no idea....

also any ideas on integral of xsecxtanx...

normally questions are with product of 2 x's but this has 3...

 

[Captain pi]

int{tanⁿx dx} = int{tan^n-2x * tan²x dx}
= int{tan^n-2x * (sec²x - 1) dx}
= int{tan^n-2x * sec²x dx} - int{tan^n-2x dx}
= tan^n-1x/n-1 - I_n-2 .... QED

This question is a bit tricky because you don't have to use integration by parts.

[Edit] Nice work Pi, you're too quick.

You did the hard work; I merely gave a quick overview.

 

[Ogden_Nash]

bah thanks alot guys, i was trying to use tanx and tan^(n-1)x just got introduced to these questions had no idea....

also any ideas on integral of xsecxtanx...

normally questions are with product of 2 x's but this has 3...

int{xsecxtanx dx} = xsecx - int{secx dx}
= xsecx - ln|secx + tanx| + C

Remember, derivative of secx = secxtanx and derivative of ln|secx + tanx| = secx