Find the indefinite integral. \int \frac{x^{3}-4x^{2}-4x+20}{x^{2}-5}dx

Michelle Arroyo

Michelle Arroyo

Answered question

2021-11-18

Find the indefinite integral.
x34x24x+20x25dx

Answer & Explanation

David Tyson

David Tyson

Beginner2021-11-19Added 19 answers

Step 1
find the indefinite integral.
x34x24x+20x25dx
Step 2
I=x34x24x+20x25dx
=(xx25+x4)dx
=xx25dx+xdx4dx
=12log(x25)+x224x+c
Ruth Phillips

Ruth Phillips

Beginner2021-11-20Added 18 answers

Step 1: Polynomial Division
x4+xx25dx
Step 2: Use Sum Rule: f(x)+g(x)dx=f(x)dx+g(x)dx.
x4dx+xx25dx
Step 3: Use Power Rule: xndx=xn+1n+1+C.
x224x+xx25dx
Step 4: Use Integration by Substitution on xx25dx.
Let u=x25,du=2xdx, then xdx=12du
Step 5: Using u and du above, rewrite xx25dx.
12udu
Step 6: Use Constant Factor Rule: cf(x)dx=cf(x)dx.
121udu
Step 7: The derivative of lnx is 1x.
lnu2
Step 8: Substitute u=x25 back into the original integral.
ln(x25)2
Step 9: Rewrite the integral with the completed substitution.
x224x+ln(x25)2
Step 10: Add constant.
x224x+ln(x25)2+C

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