Integrate e^[a*sqrt(b^2+x^2)]

abumofeed

I can integrate the first part alone, i.e. ∫e^a*√(b2+x2) and the result is [√(b²+x²)/a - 1/a²] e^a*√(b2+x2).

For the full integral, i.e. ∫e^a*√(b2+x2) * e^c*x I tried using integration by parts but it does not work since the exponent of the first e is not simply x.

the second exponent was originally a cos but that is still much harder.

Any ideas?

abumofeed

Strange, I changed the title but it looks like it did not.

SteamKing

It also helps to know with respect to which variable the integration is to be done.

abumofeed

wrt x.

LastTimelord

Let's start by taking all of the constants out of the integral:

e^(a + c)INT[e^sqrt(b^2+x^2)*x,dx]

Now we need to expand out the x to "sqrt(b^2 + x^2)(x/sqrt(b^2 + x^2))". The reason for this will become apparent.

e^(a + c)INT[e^sqrt(b^2+x^2)*sqrt(b^2 + x^2)(x/sqrt(b^2 + x^2)),dx]

Now we use U substitution where:

u = sqrt(b^2 + x^2)
du/dx = x/sqrt(b^2 + x^2)

So:

e^(a + c)INT[ue^u,du]

Now we integrate by parts:

e^(a + c)(ue^u + INT[e^u,du])

And solve:

e^(a + c)(ue^u + e^u)

Simplify:

e^(a + c)((u + 1)e^u)
(u + 1)e^(u + a + c)

Un-substitute the u:

(sqrt(b^2 + x^2) + 1)e^(sqrt(b^2 + x^2) + a + c)

There may be simplifying to be done, but you should be able to manage that. Please check over my work to see if there is anything that looks wrong before trusting me, sometimes I confuse myself when I need to format things like this with my keyboard.

abumofeed

You cannot take e^c out. C is multiplied by the square root, not added. Same for e^a.

LastTimelord

You can take out any coefficients. If it were added, it would need to have an x multiplied by it, but if the equation is a monomial, you can take any constant coefficients out of the integral, without affecting the answer.

abumofeed

see the difference:

LastTimelord

It's e^c*x, so it's logical to interpret it as "(e^c)*x", instead of what you seem to be doing: "e^(c*x), but I do now see what I did wrong: For some reason I forgot my exponent rules and thought "a^(bc) = (a^b)(a^c), when indeed it does not. Sorry, I may have been previously misunderstanding your point.

This is the correct solution:

INT[e^(a*sqrt(b^2+x^2)) * e^c*x,dx]

taking out constants, this time correctly:

(e^c)INT[e^(a*sqrt(b^2+x^2)) *x,dx]

Expanding x:

(e^c)INT[e^(a*sqrt(b^2+x^2))*(sqrt(b^2 + x^2)/a)*(ax/sqrt(b^2 + a^2)),dx]

U substitution:

u = a*sqrt(b^2+x^2)
u/a^2 = sqrt(b^2+x^2)/a
du/dx = ax/sqrt(b^2 + a^2)
du = ax/sqrt(b^2 + a^2)dx

(e^c)INT[e^(u)*(u/a^2),du]

Taking out constants:

(e^c)/(a^2)INT[u*e^(u),du]

Integration by parts:

(e^c)/(a^2)(ue^u + INT[e^u,du])

And evaluate the integral:

(e^c)/(a^2)(ue^u + e^u) + C

Un-substitute:

(e^c)/(a^2)((a*sqrt(b^2+x^2))e^(a*sqrt(b^2+x^2)) + e^(a*sqrt(b^2+x^2))) + C

Simplify:

(e^c)/(a^2)(a*sqrt(b^2+x^2) + 1)e^(a*sqrt(b^2+x^2)) + C

There may be more simplifying to do, but I won't get into that.

abumofeed

Thank you but I want e to the power of c*x.

But anyway thank you indeed.

JJacquelin

Hi !
Integral of exp[a*sqrt(b^2+x^2)]*exp(c*x)*dx
this integral cannot be expressed in terms of a finite number of standard functions, but in terms of tough series of hyperbolic functions. See attachment.
The formal result will be so ardeous that I doubt that it could be useful for further calculus in practice.
So, I suggest to use numerical methods instead of analytical.

Attached Thumbnails

 

abumofeed

Thanks indeed!

Yes I tried a lot but have not find any good result and you confirmed this. Thanks for your effort, I really appreciate your time! Thanks again.