Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2007-08-14 21:55:02

Hastega
Member
Registered: 2007-06-20
Posts: 7

Help! Integral Calculus

Can someone help me with these two problems?

bonga.jpg

Offline

#2 2007-08-15 00:05:41

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: Help! Integral Calculus

According to Wikipedia table of integrals:

but it didnt have anything for cotangent.


The Beginning Of All Things To End.
The End Of All Things To Come.

Offline

#3 2007-08-15 01:15:57

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Help! Integral Calculus

Possibly split the cotangent into sine and cosine, then use integration by parts?  Just a guess.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Offline

#4 2007-08-15 04:09:57

HallsofIvy
Guest

Re: Help! Integral Calculus

The first is an easy "standard" problem because it has sine to an ODD power.

  Write it as int sin^6(x) (sin(x)dx) = int (1- cos^2(x)^3 (sin(x)dx).  Now let u= cos(x) so that du= -sin(x)dx and we have -int (1- u^2)^3 du= -int (1- 3u^2+ 3u^4- u^6)du= -(u- u^3+ (3/5)u^5- (1/7)u^7)+ C= (1/7)cos^7(x)-(3/5)cos^5(x)+ cos^3(x)- cos(x)+ C.

  I'm surprised Wikipedia doesn't have such a formula for cot.  My Calculus book gives
int cot^n (x)dx= - (cot^(n-1)(x))/(n-1)- int cot^(n-2)(x)dx.

#5 2007-08-15 20:02:27

Hastega
Member
Registered: 2007-06-20
Posts: 7

Re: Help! Integral Calculus

So (1/7)cos^7(x)-(3/5)cos^5(x)+ cos^3(x)- cos(x)+ C is the answer to number 1.

I believe the formula for Cot is fddd.jpg

But i still can't answer it, I always get stuck because there not integrable.

Offline

#6 2007-08-15 20:30:31

luca-deltodesco
Member
Registered: 2006-05-05
Posts: 1,470

Re: Help! Integral Calculus

for 1 i get:





which according wolfram integrator can be simplified to:

Last edited by luca-deltodesco (2007-08-15 20:33:36)


The Beginning Of All Things To End.
The End Of All Things To Come.

Offline

#7 2007-08-16 01:09:07

gnitsuk
Member
Registered: 2006-02-09
Posts: 121

Re: Help! Integral Calculus

Problem 2:

To solve this we need only the basic result:


Which is easily obtained via the trig relation:

For then:

for n >= 2.

(Which is the formula metioned by HallsofIvy)

So we can always reduce to the case n = 1 or n = 0.

This will allow you to solve the given integral.

Last edited by gnitsuk (2007-08-20 20:57:54)

Offline

Board footer

Powered by FluxBB