Solving this definite integral using integration by parts

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  • #1
songoku
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Homework Statement
Please see below
Relevant Equations
Integration
1684334479139.png


Using integration by parts:
$$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$
$$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$
Then how to continue?

Thanks
 
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  • #2
My first approach would be work out what In+1 is and see how they relate.

Their hint of multiplying by 1 tells me that I might need to represent 1 as a fraction where numerator = denominator.
 
Last edited:
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  • #3
Hint: ##x^2 = (x^2+1) - 1##.
 
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  • #4
I understand

Thank you very much scottdave and vela
 
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