Integration with tables

[Null_]

Homework Statement

Find ∫(x^2+x+5)/√(x^2+1)dx using a table of integrals

Homework Equations

These are the forms I chose to use:
#1 ∫(u^2)/(√(u^2 + a^2))du = (u/2)*(√(a^2 + u^2)) - (a^2/2)(ln(u+√(a^2 + u^2) + C

and

#3 ∫(du)/(√(a^2 + u^2)) = ln(u + √(a^2 + u^2) +C

The Attempt at a Solution

I won't show every step, but here's what I did:

1. Split the single integral up into three separate integrals.
2. For the first, I let u=x and a=1. I then plugged into the formula and got...
(x/2)*√(1+x^2)-(1/2)*ln(x+√(1+x^2)) + C
3. Solve the second with just a u substitution...
= √(x^2 + 1) + C
4. Solve the third with a table... I got
= 3*√(1+x^2) + 9*ln(x+ √(1+x^2))/2 + C


I'm pretty sure that I used the correct integrations from the table, and I can't seem to see where I went wrong. I know the final answer just by using Maple. I think I can also solve it without using a table. I would use trig to do that, but that's not what this homework is testing us on.

I sincerely apologize for my lack of LaTex use...
If you want to know more of the steps I took, just say so and I will post a picture of my work.

Thanks

 

[SammyS]

Homework Statement

Find the integral of (x^2+x+5)/sqrt(x^2+1)dx using a table of integrals

Homework Equations

These are the forms I chose to use:
#1 Integral(u^2)/(sqrt(u^2 + a^2))du = (u/2)*(sqrt(a^2 + u^2)) - (a^2/2)(ln(u+sqrt(a^2 + u^2) + C

and

#3 Integral (du)/(sqrt(a^2 + u^2)) = ln(u + sqrt(a^2 + u^2) +C

The Attempt at a Solution

I won't show every step, but here's what I did:

1. Split the single integral up into three separate integrals.

What were the three parts before you started into the integrating?

2. For the first, I let u=x and a=1. I then plugged into the formula and got...
(x/2)*sqrt(1+x^2)-(1/2)*ln(x+sqrt(1+x^2)) + C
3. Solve the second with just a u substitution...
= sqrt(x^2 + 1) + C
4. Solve the third with a table... I got
= 3*sqrt(1+x^2) + 9*ln(x+ sqrt(1+x^2))/2 + C


I'm pretty sure that I used the correct integrations from the table, and I can't seem to see where I went wrong. I know the final answer just by using Maple. I think I can also solve it without using a table. I would use trig to do that, but that's not what this homework is testing us on.

I sincerely apologize for my lack of LaTex use.

If you want to know more of the steps I took, just say so and I will post them.

Thanks

I would try splitting the integrand up as follows:

$$\frac{x^2+x+5}{\sqrt{x^2+1}}=\frac{x^2+1+x+4}{\sqrt{x^2+1}}$$

$$=\frac{x^2+1}{\sqrt{x^2+1}}+\frac{x}{\sqrt{x^2+1}}+\frac{4}{\sqrt{x^2+1}}$$
​

 

[Null_]

I split it into:
∫ (x^2)/(√(x^2+1)) dx + ∫ x/(√(x^2+1)) dx + 5∫dx/√(x^2+1)

While I understand why you split it up as you did, the first part is not listed in the tables of my book (http://teachers.sduhsd.net/abrown/Classes/CalculusC/IntegralTablesStewart.pdf [Broken] ).

 

[dextercioby]

For the records, the problem is more difficult than simply coputing the integral all by itself by applying the substitution $x=\sinh t$. It makes no sense to use integral tables when the results can be reached with minimum math knowledge...But, hey, not all teachers are smart.

 

[Null_]

Yeah, I can solve it without using integration with tables by using the substitution you suggested. It was due this morning, so I've already gotten the answer "right," but I'd still really like to know how to solve it using tables and why my choices didn't work.

 

[SammyS]

I split it into:
∫ (x^2)/(√(x^2+1)) dx + ∫ x/(√(x^2+1)) dx + 5∫dx/√(x^2+1)

While I understand why you split it up as you did, the first part is not listed in the tables of my book (http://teachers.sduhsd.net/abrown/Classes/CalculusC/IntegralTablesStewart.pdf [Broken] ).

Thanks for including a link to those integral tables.

I left $$\frac{x^2+1}{\sqrt{x^2+1}}$$ unsimplified. It is, of course, equal to $$\sqrt{x^2+1}$$, and form 21 works for this. The way you broke up the integrand should have worked fine.

I'm curious as to how you got your result for $$\int{{1}\over{\sqrt{x^2+1}}}\,dx\,.$$


BTW, What is the answer you were trying to match? - the one from Maple.

 

[Null_]

The answer for the problem (it is an online problem and the key is already up) is:
(1/2)*x*√(x^2+1) + √(x^2+1) + (9/2)*ln(x+√(x^2+1))

I'm still having a hard time seeing why my answer wasn't right. I've re-done it five times and ended up with the same answer.

The answer from maple uses sinh^-1(x), so it must have done the substitution method.

 

[SammyS]

The answer for the problem (it is an online problem and the key is already up) is:
(1/2)*x*√(x^2+1) + √(x^2+1) + (9/2)*ln(x+√(x^2+1))

I get this answer using the tables.

I'm still having a hard time seeing why my answer wasn't right. I've re-done it five times and ended up with the same answer.

The answer from maple uses sinh^-1(x), so it must have done the substitution method.

BTW: $$sinh^{-1}(x)=\ln\left(x+\sqrt{x^2+1}\right)$$

 

[SammyS]

∫(du)/(√(a^2 + u^2)) = ln(u + √(a^2 + u^2) +C

So, $$\int \frac{dx}{\sqrt{x^2+1}}=\ln\left(x+\sqrt{x^2+1}\right)+C\,.$$

How did you get the following?

4. Solve the third with a table... I got
= 3*sqrt(1+x^2) + 9*ln(x+ sqrt(1+x^2))/2 + C