Substitution Integral for ∫1/(x*sqrt(x^2-1)) using Simple Substitution Method

[Jalo]

Homework Statement

Find the following integral

∫1/(x*sqrt(x^2-1) dx

Homework Equations

The Attempt at a Solution

I've decided to use the substitution:

x = sec u
dx = sec u * tan u du

Substituting on the integral I got:

∫sec(u)*tan(u) / [sec u * sqrt((sec^2(u)-1))] du

Since 1+tan^2(u) = sec^2(u) the integral simplifies to
∫ sec(u)*tan(u) / [sec(u)*tan(u)] du = ∫ du = u + c = sec(u) + c, c being an arbitrary constant.

The answer on the solutions is given by the substitution

u = sqrt(x^2-1)

Is my answer wrong? Because it seems way simplier this way, and I don't see nothing wrong with the substitution...

If anyone could help me I'd appreciate!

Thanks.

 

[Dick]

Homework Statement

Find the following integral

∫1/(x*sqrt(x^2-1) dx

Homework Equations

The Attempt at a Solution

I've decided to use the substitution:

x = sec u
dx = sec u * tan u du

Substituting on the integral I got:

∫sec(u)*tan(u) / [sec u * sqrt((sec^2(u)-1))] du

Since 1+tan^2(u) = sec^2(u) the integral simplifies to
∫ sec(u)*tan(u) / [sec(u)*tan(u)] du = ∫ du = u + c = sec(u) + c, c being an arbitrary constant.

The answer on the solutions is given by the substitution

u = sqrt(x^2-1)

Is my answer wrong? Because it seems way simplier this way, and I don't see nothing wrong with the substitution...

If anyone could help me I'd appreciate!

Thanks.

If the integral comes out to u+c, and you want to express it in terms of x, u=arcsec(x). u isn't equal to sec(u). And the answer on the solutions doesn't work. Test it by taking the derivative. arcsec(x)+c does work.

 

[Jalo]

If the integral comes out to u+c, and you want to express it in terms of x, u=arcsec(x). u isn't equal to sec(u). And the answer on the solutions doesn't work. Test it by taking the derivative. arcsec(x)+c does work.

I'm sorry but I'm a little confused. The result is obviously incorrect, but I don't think i did any wrong assumption as I was solving the integral. All the mathematical steps appear to be correct, including the substitution. How can it be wrong then?

I solved it with the substitution from the solutions and I got to the result arctan(sqrt(x^2-1)), which is different from arcsin but is correct (I confirmed it with Matlab).

 

[Mark44]

I'm sorry but I'm a little confused. The result is obviously incorrect, but I don't think i did any wrong assumption as I was solving the integral. All the mathematical steps appear to be correct, including the substitution. How can it be wrong then?

Your work was fine up until you undid your substituion. If x = sec(u), then u = sec^-1(x) or arcsec(x).

When you undid your substitution, you replaced u with sec(u), which is incorrect. That's what Dick was saying.

I solved it with the substitution from the solutions and I got to the result arctan(sqrt(x^2-1)), which is different from arcsin but is correct (I confirmed it with Matlab).

 

[Dick]

I'm sorry but I'm a little confused. The result is obviously incorrect, but I don't think i did any wrong assumption as I was solving the integral. All the mathematical steps appear to be correct, including the substitution. How can it be wrong then?

I solved it with the substitution from the solutions and I got to the result arctan(sqrt(x^2-1)), which is different from arcsin but is correct (I confirmed it with Matlab).

Your solution is not wrong. You got u+c. Since x=sec(u), u=arcsec(x). That is also correct. I read your post wrong. I thought you said the solution given was sqrt(x^2-1), but you didn't say that, you just said the subsitution was u=sqrt(x^2-1). Sorry!

 

[Jalo]

Ooh, I see it now... Thank you for your answer!