Thursday, August 15, 2019

calculus - Simpler way to compute a definite integral without resorting to partial fractions?


I found the method of partial fractions very laborious to solve this definite integral : $$\int_0^\infty \frac{\sqrt[3]{x}}{1 + x^2}\,dx$$


Is there a simpler way to do this ?


Answer



Perhaps this is simpler.



Make the substitution $\displaystyle x^{2/3} = t$. Giving us


$\displaystyle \frac{2 x^{1/3}}{3 x^{2/3}} dx = dt$, i.e $\displaystyle x^{1/3} dx = \frac{3}{2} t dt$


This gives us that the integral is


$$I = \frac{3}{2} \int_{0}^{\infty} \frac{t}{1 + t^3} \ \text{d}t$$


Now make the substitution $t = \frac{1}{z}$ to get


$$I = \frac{3}{2} \int_{0}^{\infty} \frac{1}{1 + t^3} \ \text{d}t$$


Add them up, cancel the $\displaystyle 1+t$, write the denominator ($\displaystyle t^2 - t + 1$) as $\displaystyle (t+a)^2 + b^2$ and get the answer.


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