Evaluate the integral. \int \frac{x^{2}-5x+16}{(2x+1)(x-2)^{2}}dx

eiraszero11cu

eiraszero11cu

Answered question

2021-12-27

Evaluate the integral.
x25x+16(2x+1)(x2)2dx

Answer & Explanation

Toni Scott

Toni Scott

Beginner2021-12-28Added 32 answers

Step 1
The integral given is, x25x+16(2x+1)(x2)2dx.
After partial fraction decomposition,
x25x+16(2x+1)(x2)2dx=32x+1dx1x2dx+2(x2)2dx.
Solve first part of integral ,
Substitute u2x+1,du=2dx.
32x+1dx=32duu
=32lnu+c1.
Step 2
After replace u with 2x+1,
32x+1dx=32ln(2x+1)+c1...(1)
Solving the second part of integral,
Substitute u\Rightarrow x-2, du=dx.
dxx2=duu
=lnu+c2.
Replace u with x-2,
dxx2=ln(x2)+c2...(2)
Step 3
Solving the third part of the integral,
Substitute u  x-2, du=dx.
dx(x2)2=duu2
=1u+c3.
Replace u with x-2,
dx(x2)2=1x2+c3...(3)
Step 4
The final solution of the integration becomes, adding equation 1 , 2 and 3,
x25x+16(2x+1)(x2)2dx=32x+1dx1x2dx+2(x2)2dx.

Debbie Moore

Debbie Moore

Beginner2021-12-29Added 43 answers

x25x+16(2x+1)×(x2)2dx
32x+11x2+2(x2)2dx
32x+1dx1x2dx+2(x2)2dx
32×ln(|2x+1|)ln(|x2|)2x2
Add C
Solution
32×ln(|2x+1|)ln(|x2|)2x2+C
karton

karton

Expert2022-01-04Added 613 answers

x25x+16(x2)2(2x+1)dx=(32x+11x2+2(x2)2)dx=312x+1dx1x2dx+21(x2)2dx12x+1dx=121udu1udu=ln(u)121udu=ln(u)2=ln(2x+1)21x2dx=1udu=ln(u)=ln(x2)1(x2)2dx=1u2du=1u=1x2312x+1dx1x2dx+21(x2)2dx=3ln(2x+1)22x2ln(x2)x25x+16(x2)2(2x+1)dx=3ln(|2x+1|)22x2ln(|x2|)+C

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?