A Reduction Formula and a Special MethodDate: 1/23/96 at 10:47:40 From: Anonymous Subject: MATH Hello. I'm Burcak and I'm sending you two questions: 1 ) F(X)=SEC(4X)^5 I'm asking for the integration of F(X). ('^5' is used for the fifth power of F(X) ) 2) F(X)=(2SIN(X)+3COS(X))/(3SIN(X)+2COS(X)) Again, I'm asking for the integration of F(X). Thanks! Date: 7/22/96 at 22:0:42 From: Doctor Jerry Subject: Re: MATH A convenient formula (called a reduction formula) for solving questions like (1) is: int(sec^n x dx)=(1/(n-1)) sec^(n-2) x tan(x) + ((n-2)/(n-1)) int( sec^(n-2) x dx. n is a positive number, greater than or equal to 3. The idea is to use the formula repeatedly, until the n-2 becomes either 1 or 2. The integral of secant to the first or second powers is a standard, known formula. First, however, make the substitution 4x=w in the original integral. Question (2) is more interesting. Although it is a rational function of sin x and cos x and can be solved by the well known substitution u=tan(x/2), it can also be solved by a special method. First, note that (2sin x+3cos x)/(3sin x+2cos x)+K = ((2+3K)sin x+(3+2K)cos x)/(3sin x+2cos x). So that the numerator is a numerical multiple of the derivative of the denominator, we try to find L so that 2+3K=-2L and 3+2K=3L. The solution of these equations is K = -12/13 and L = 5/13. From this it follows that the integral of your F(X) is (12X)/13+ 5LN(2COS X+3SIN x)/13. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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