IntegrationDate: 5/29/96 at 10:56:45 From: Anonymous Subject: Integration Integral of (cos(x).x(cub).sinh(x)square).dx What does it mean and what's the answer? Date: 6/28/96 at 14:27:3 From: Doctor Jerry Subject: Re: Integration If you look in your calculus book, you will find integrals of the form int(exp(ax) cos(bx) dx) in the integration by parts section. You can then do integration by parts on int(x^3 (exp(ax) cos(bx) dx) by letting u=x^3. To obtain the exponential form, replace sinh(x) by its equal, (exp(x)-exp(-x))/2. Here's an answer, obtained from Mathematica. You may notice that Mathematica has used various hyperbolic trig function identities to re-express the result (note, for example, cosh(3x)). (5625*x*Cos[x]*Cosh[x] - 1875*x^3*Cos[x]*Cosh[x] + 135*x*Cos[x]* Cosh[3*x] + 375*x^3*Cos[x]*Cosh[3*x] + 5625*x^2*Cosh[x]*Sin[x] - 72*Cosh[3*x]* Sin[x] - 225*x^2*Cosh[3*x]*Sin[x] - 5625*Cos[x]*Sinh[x] - 5625*x*Sin[x]* Sinh[x] - 1875*x^3*Sin[x]*Sinh[x] - 21*Cos[x]*Sinh[3*x] - 300*x^2*Cos[x]* Sinh[3*x] +195*x*Sin[x]*Sinh[3*x] + 125*x^3*Sin[x]*Sinh[3*x])/5000 As to what this integral means, many answers can be given. Most directly, it is a function whose derivative is x^3 cos x sinh x. If you had given a definite integral, then the numerical answer would be the area under the curve x^3 cos x sinh x, between the two limits. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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