- #1
8614smith
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Homework Statement
sketch the region of integration, and evaluate the integral by choosing the best order of integration
[tex]\int^{8}_{0}\int^{2}_{x^{1/3}}\frac{dydx}{y^{4}+1}[/tex]
Homework Equations
integration by parts
The Attempt at a Solution
after sketching the graph and changing the limits I've got to:
[tex]\int^{2}_{0}\int^{y^3}_{0}\frac{dydx}{y^{4}+1}[/tex]
integrating:
[tex]\int^{2}_{0}\left[\frac{x}{y^{4}+1}\right]^{y^3}_{0}dy=\int^{2}_{0}\frac{y^3}{{y^4}+1}dy[/tex]
[tex]u = {y^3}[/tex]
[tex]u' = 3{y^2}[/tex]
[tex]v = ln({y^4}+1)[/tex]
[tex]v' = \frac{1}{{y^4}+1}[/tex]
[tex]\left[{y^3}ln({y^4}+1)-\int3{y^2}ln({y^4}+1)\right][/tex]
[tex]u = ln({y^4}+1)[/tex]
[tex]u' = \frac{1}{{y^4}+1}[/tex]
[tex]v = {y^3}[/tex]
[tex]v' = 3{y^2}[/tex]
[tex] = {y^3}ln({y^4}+1)-\int{y^3}\frac{1}{{y^4}+1}[/tex]
And this is where i get stuck as its just an infinite loop of integration by parts, any ideas?