Kelumptus 2 years ago Solution to the following integral (in post below)

1. Kelumptus

$\int\limits_{}^{} \frac{t-1}{t^{2}+4t+3} dt$ I'm kind of stuck on this one, i've tried a few different methods of u substitution but can't solve it.

2. Callisto

Does partial fraction help?

3. Kelumptus

Yeah i think that might be the answer. I have only done partial fractions in the form where there is a coefficient and a variable however. Not when there are two terms in the numerator.

4. Callisto

Hmmm... $\frac{t-1}{t^{2}+4t+3}=\frac{t-1}{(t+1)(t+3)} = \frac{A}{t+1}+\frac{B}{t+3}$ So, A(t+3) + B(t+1) = t-1 Comparing the coefficients, A+B = 1 3A +B = -1 Something like this?!

5. Kelumptus

Yep... i think i jumped the gun when i gave up on trying to solve with partial fractions. I just revised my notes and i'll have a go using partial fractions now. I'll let you know how i go.

6. Callisto

Okay!!~

7. Kelumptus

Awesome, partial fractions was the key. Thanks for that =)

8. Callisto

You're welcome :)

9. sehanhasan

Use partial fraction method..

10. shubhamsrg

or you may use partial fractions alternatively, what i mean is t-1 / t^2 +4t +3 = 1/2 ( 2t - 2)/(t^2 +4t +3) =1/2 ( 2t +4 -6)/(t^2 +4t +3) = 1/2 ( 2t +4)/(t^2 +4t +3) - 1/2 (6/(t^2 +4t +3)) integrate separately, for 1st part, if you let denominator =z,. you directly get numerator = dz .. we are concerned with integrating 1/(t^2 +4t +3) that's also simple, we see we have 1/(t+1)(t+3) = 1/2 (2 / (t+1)(t+3)) =1/2 ( t+3 - (t+1))/(t+1)(t+3)) =>now when you separate out, you'll directly be able to integrate ..hope that helps.. note that i've used partial fractions only,but in a different manner..