Asked by hewhomustbenamed82
Show that the following integral is convergent,
Integral, it goes from 0 to infinity dx/(sqrt(x) + x^2)
Integral, it goes from 0 to infinity dx/(sqrt(x) + x^2)
Answers
Answered by
Steve
Let u^2 = x
2u du = dx
and now the integral becomes
∫ 2u/(u+u^4) du = 2∫du/(1+u^3) = 2∫du/((u+1)(u^2-u+1)
= 2/3 ∫(1/(u+1) - (u-2)/(u^2-u+1)) du
= 2/3 ∫ 1/(u+1) - (1/2)(2u-1)/(u^2-u+1) + (3/2)/(u^2-u+1) du
= 2/3 ∫ 1/(u+1) - (1/2)(2u-1)/(u^2-u+1) + (3/2)/((u-1/2)^2 + 3/4) du
Now you can see how to do each of those, using log and arctan ... right?
check my math on all those fractions
2u du = dx
and now the integral becomes
∫ 2u/(u+u^4) du = 2∫du/(1+u^3) = 2∫du/((u+1)(u^2-u+1)
= 2/3 ∫(1/(u+1) - (u-2)/(u^2-u+1)) du
= 2/3 ∫ 1/(u+1) - (1/2)(2u-1)/(u^2-u+1) + (3/2)/(u^2-u+1) du
= 2/3 ∫ 1/(u+1) - (1/2)(2u-1)/(u^2-u+1) + (3/2)/((u-1/2)^2 + 3/4) du
Now you can see how to do each of those, using log and arctan ... right?
check my math on all those fractions
There are no AI answers yet. The ability to request AI answers is coming soon!