I'm searching the antiderivative of rational functions:1)\int\frac{1+x}{\sqrt{2x+1}}dxF

Cosmo Truong

Cosmo Truong

Answered question

2022-02-17

I'm searching the antiderivative of rational functions:
1)1+x2x+1dx
For this one we have t=2x+1 then st=12x+2dx but then I do not see the way to compute the antiderivative.
Same thing for 2)1x+x+1
What are the methods to find antiderivative of rational functions like these ones?
Thank you For (1): Try the substitution u=2x+1. Then du=2dx12du=dx, and
x=12(u1) x+1=12(u1)+1=12(u+1)
Then your integral becomes
1+x2x+1dx=14u+1u12du=14
(u12+u12)du

Answer & Explanation

Cicolinif73

Cicolinif73

Beginner2022-02-18Added 7 answers

For (1): Try the substitution u=2x+1. Then du=2dx12du=dx, and
x=12(u1) x+1=12(u1)+1=12(u+1)
Then your integral becomes
1+x2x+1dx=14u+1u12du=14
(u12+u12)du

klimatskav41

klimatskav41

Beginner2022-02-19Added 4 answers

Using your substitution, t=2x+1, so dt=2dx22x+1=dx2x+1. On the other hand, x=12(t21), so the integral becomes
(1+x)dx2x+1=(1+12(t21))dt=12(1+t2)dt,
which Im

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?