anonymous 4 years ago Integral Question

1. anonymous

f(x)=\int_1^{3x^5} \ln(t^2+1) \, dt

2. anonymous

$f(x)=\int_1^{3x^5} \ln(t^2+1) \, dt$like that?

3. anonymous

is the question find the derivative?

4. anonymous

yes it is,

5. anonymous

how'd i guess?

6. anonymous

too hard to integrate

7. anonymous

use the chain rule. the derivative of the integral is the integrand, so replace t by $3x^5$ and then multiply the answer by $15x^4$

8. anonymous

yeah im having trouble with these. Ill try it sat!

9. anonymous

$f'(x)=\ln((3x^5)^2+1)\times 15x^4$

10. anonymous

i should have let you do it on your own, but once you see it you should realize how easy it is. it is clear what i did yes?

11. anonymous

if you need a word of explanation let me know

12. anonymous

i understand chain rule for ln (function within function). where did the 15x4 come from?

13. anonymous

nvm 3x^5 derivative

14. anonymous

15x^4 \ln(9x^{10}+1) ?

15. anonymous

yes

16. anonymous

it is the chain rule and you have a composite function. don't forget that $F(x)=\int_a^x f(t)dt$ is a function of "x" so if you do not have an x in the upper limit, but rather something else, you have a composition like $F(g(x))=\int_a^{g(x)}f(t)dt$ and so you need the chain rule