How Can You Solve This Challenging Integral Evaluation Problem?

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SUMMARY

The integral evaluation problem discussed involves the integral \(\int\frac{\arctan(x)dx}{(1+x^2)^{\frac{3}{2}}}\). The solution utilizes the substitution \(x=\tan(u)\), leading to the transformation \(\frac{dx}{du}=\frac{1}{\cos^{2}(u)}\) and subsequently simplifying the integral to \(\int u \cos(u) du\). This approach demonstrates that the problem, initially perceived as challenging, can be effectively solved with the right substitution technique.

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hadi amiri 4
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Evaluate
\int\frac{arctan(x)dx}{(1+x^2)^\frac{3}{2}}
 
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Make the substitution:
x=\tan(u),\to\frac{dx}{du}=\frac{1}{\cos^{2}u}
Thus, we get:
dx=\frac{du}{\cos^{2}(u)}
and insertion in your integral yields:
\int\frac{arctan(x)}{(1+x^{2})^{\frac{3}{2}}}=\int{u}\cos(u)du
 


your solution seems nice
honestly i thought it is a hard one,becouse i picked it form "A coures of pure mathematics"
 

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