## mony01 2 years ago Anyone know how to do this integral?

1. mony01

$\int\limits \frac{ x ^{2} +x+1}{ (x ^{2}+1)^{2} }dx$

2. anonymous

Play around with the fraction.

3. anonymous

Maybe partial fraction decomposition

4. myininaya

did you try a trig sub i think that would work just fine

5. mony01

you think i can try sin?

6. myininaya

I think you can try tan

7. myininaya

the hint was the x^2+1 on bottom

8. myininaya

tan^2(theta)+1=sec^2(theta)

9. mony01

would the set up be integral x^2+x+1/sex^2 theta

10. myininaya

well you have to replace all the x's and the dx

11. myininaya

and also you are leaving off the square on bottom

12. myininaya

$x=\tan(\theta) => dx=\sec^2(\theta) d \theta$ Replace all the x's with tan(theta) Replace the dx with sec^2(theta) d theta

13. mony01

is it integral tan^2theta+tan theta+1/sec^2 theta d (theta)

14. myininaya

If I think what you wrote is what I think then yes. You mean the following I assume: integral of (tan^2(theta)+tan(theta)+1)/sec^2(theta) d(theta)

15. myininaya

$\int\limits_{}^{}\frac{\tan^2(\theta)}{\sec^2(\theta)} d \theta +\int\limits_{}^{}\frac{\tan(\theta)}{\sec^2(\theta)} d \theta +\int\limits_{}^{} \frac{1}{\sec^2(\theta) }d \theta$

16. myininaya

Look at them three separately

17. gorv

$\int\limits_{}^{}(\frac{ x^2+1 }{ (x^2+1)^2 } +\frac{ x }{ (x^2+1)^2 })*dx$

18. gorv

$\int\limits_{}^{}\frac{ dx }{ x^2+1 } +\int\limits_{}^{}\frac{ x }{ (x^2+1)^2 }$

19. gorv

first one is standard formula

20. mony01

how can i figure out the answer?

21. gorv

$\int\limits_{}^{}\frac{ x }{ (x^2+1)^2 } *dx$ for this x^2+1=t 2xdx=dt xdx=dt/2

22. gorv

$\int\limits_{}^{} \frac{ dt }{ 2*t^2 }$

23. gorv

$\int\limits_{}^{}\frac{ dx }{ x^2+1 } +\int\limits_{}^{}\frac{ dt }{ 2t^2 }$

24. gorv

can u solve it @mony01

25. mony01

is it sec^2 (theta)/2(x^2+1)^2

26. myininaya

I do like gorv's way. But either way is fine. gorv's is simpler though.

27. gorv

actual its less calculative...both ways anns will be same

28. mony01

how can i solve the rest of it?

29. myininaya

What question on what part do you have?

30. mony01

$\int\limits \frac{ dx }{ x ^{2}+1}+\int\limits \frac{ dt }{ 2t ^{2} }$

31. myininaya

the first integral is just remembering that it is acran(x) the second one 1/2 is a constant multiple and you should know how to integrate t^(-2) at this point in calculus

32. mony01