Evaluate the integral. \int \frac{x^{4}+3x^{2}+1}{x^{5}+5x^{3}+5x}dx

untchick04tm

untchick04tm

Answered question

2021-12-09

Evaluate the integral.
x4+3x2+1x5+5x3+5xdx

Answer & Explanation

Timothy Wolff

Timothy Wolff

Beginner2021-12-10Added 26 answers

Step 1
Given: x4+3x2+1x5+5x3+5xdx
for evaluating given integral we substitute
x5+5x3+5x=t
now differentiating both side with respect to x
5x4+5(3x2)+5=dtdx   (ddx(kxn)=k(nxn1))
5(x4+3x2+1)=dtdx
(x4+3x2+1)dx=dt5
now substitute (x4+3x2+1)dx with dt5 and (x5+5x3+5x) with t in given integral
Step 2
so,
x4+3x2+1x5+5x3+5xdx=dt5
=15dtt   (dxx=lnx+c)
=lnt5+c
now replacing t with (x5+5x3+5x)
so, given integral is ln(x5+5x3+5x)5+c
hence, given integral is equal to ln(x5+5x3+5x)5+c.

Deufemiak7

Deufemiak7

Beginner2021-12-11Added 34 answers

x4+3x2+1x5+5x3+5xdx
Transform the expression
15tdt
Use properties of integrals
15×1tdt
Evaluate the integral
15×ln(|t|)
Substitute back
15×ln(|x5+5x3+5x|)
Add CR
Solution
15×ln(|x5+5x3+5x|)+C

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