# Substitution process

1. Dec 8, 2007

### rayray19

1. The problem statement, all variables and given/known data

1) antiderivative of ((t^2)+2)/((t^3)+6t+3) dt

2) antiderivative of r(sqrt((r^2)+2))dr

2. Relevant equations

3. The attempt at a solution

#2 let u = r^2 + 2

du/dr = 2r

du = 2rdr?? i dont knoww!!

2. Dec 8, 2007

### rocomath

$$\int\frac{t^2 +2}{t^3 +6t +3}dt$$

$$\int r\sqrt{r^2+2}dr$$

correct?

3. Dec 8, 2007

### rayray19

yes, that is correct

4. Dec 8, 2007

### cristo

Staff Emeritus
For the first one, try partial fractions (probably).

For the second, your substitution seems promising. You will have the integral $$\int r u^{1/2}\frac{du}{2r}=\int \frac{1}{2}u^{1/2}du$$

Last edited: Dec 8, 2007
5. Dec 8, 2007

### rocomath

well for both, all you do is a u-substitution

so let's work 2

$$\int r\sqrt{r^2+2}dr$$

$$u=r^2 +2$$
$$du=2rdr \rightarrow \frac{1}{2}du=rdr$$

Rearranging your integral, do you notice that your derivative shows up in your original integrand? by that happening, you can take it out of your integral.

$$\int\sqrt{r^2 +1} rdr$$

$$\frac{1}{2}\int\sqrt{u}du$$

6. Dec 8, 2007

### rayray19

for my answer to #2 i got (1/3)((r^2)+2)^(3/2) + c

is that correct??

7. Dec 8, 2007

### rocomath

correct, now your first one works out the same way, all you have to do is factor our a common term from the derivative of your u-sub.

8. Dec 8, 2007

### rayray19

i factored out a 3 but now im lost at finishing it up

9. Dec 8, 2007

### rayray19

i got 3 times the antiderivative of du/u dt.. i dont think thats right though

10. Dec 8, 2007

### rocomath

$$u=t^3 +6t +3$$
$$du=3(t^2 + 1)dt \rightarrow \frac{1}{3}du=(t^2 +1)dt$$

just replace what you have with your u-sub and derivative of your u-sub.

Last edited: Dec 8, 2007
11. Dec 8, 2007

### rocomath

What you have to do is, divide by that 3 so that it becomes the constant for your substituted Integral.

Last edited: Dec 8, 2007
12. Dec 8, 2007

### rayray19

im sorry im stuck, what do i do after i find the du=3((t^2) +2)) dt

13. Dec 8, 2007

### rocomath

Last edited: Dec 8, 2007
14. Dec 8, 2007

### rayray19

so is the answer (1/3)((t^2)+2) +c ???

15. Dec 8, 2007

### rocomath

Unfortunately, no. Have you learned about the Integral of Ln? (Natural Log)

16. Dec 8, 2007

### rayray19

yea it becomes 1/x doesnt iit

17. Dec 8, 2007

### rocomath

correct, so when we complete all our substitutions, we end up with:

$$\frac{1}{3}\int\frac{1}{u}du$$

so now take the Integral of that and just resubstitute.

18. Dec 8, 2007

### rayray19

i think i have it,, (1/3)ln(t^3 + 6t + 3) ?????

19. Dec 8, 2007

### rocomath

yes, but with + C

20. Dec 8, 2007

### cristo

Staff Emeritus
It helps if you post your working at each stage, instead of simply posting what you get as the answer. This not only helps the person checking your work, but also helps you in that you organise your thoughts into a logical progression through the problem.