Homework Help

What is integral (0 to 1) x^3/(x^4+1) ?  

user profile pic

pillbill | Student, Undergraduate | eNoter

Posted July 1, 2013 at 1:45 PM via web

dislike 1 like

What is integral (0 to 1) x^3/(x^4+1) ?

 

1 Answer | Add Yours

user profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 1, 2013 at 1:52 PM (Answer #1)

dislike 1 like

You should come up with the following substitution, such that:

`x^4 + 1 = t`

Differentiating both sides yields:

`4x^3 dx = dt => x^3 dx = (dt)/4`

Replacing the limits of integration yields:

`x = 0 => t = 1`

`x = 1 => t = 1^4 + 1 = 2`

Replacing the variable to integrand, yields:

`int_1^2 (dt)/(4t) = (1/4)int_1^2 (dt)/t`

`(1/4)int_1^2 (dt)/t = (1/4)ln t|_1^2`

Using the fundamental theorem of calculus, yields:

`(1/4)int_1^2 (dt)/t = (1/4)(ln 2 - ln 1)`

Since `ln 1 = 0` yields:

`(1/4)int_1^2 (dt)/t = (1/4)(ln 2) => (1/4)int_1^2 (dt)/t = ln 2^(1/4)`

`(1/4)int_1^2 (dt)/t = ln root(4)2`

Hence, evaluating the given definite integral, yields` int_0^1 x^3/(x^4 + 1)dx = ln root(4)2.`

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes