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### Integration

```
Date: 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/
```
Associated Topics:
High School Calculus
High School Trigonometry

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