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Jonask
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Let \[I_n=\huge\int \frac{x^n}{\sqrt{x^2+a}}\] \[n \in N,a>0,and \space I_n=(a,\infty)\]
Jonask
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Find \[I_0,I_1 ,I_n\]
Jonask
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I did the first two but three i dont understand
Jonask
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actually it says show that \[I_n=\frac{1}{n}x^{n-1}\sqrt{x^2+a}-\frac{n-1}{n}aI_{n-2}\]
Jonask
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\[I_0=\ln(x+\sqrt{x^2+a})-\ln\sqrt{a}+c,I_1=\frac{1}{2}\sqrt{x^2+a}+c\]
hartnn
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tried integrating by parts ?
Jonask
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yes\[ f=x^n,f'=nx^{n-1}\]
\[g'=\frac{1}{\sqrt{x^2+a}}.g=I_0\]
\[I_n=x^n(I_0)-\int I_0nx^{n-1}\]
Jonask
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\[I_0\] is known above
Jonask
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\[I_n=x^n\ln(x+\sqrt{x^2+a})-n\int x^{n-1}\ln(x+\sqrt{x^2+a})\]
Jonask
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using parts twice
\[\int(x^{n-1})\ln(x+\sqrt{x^2+a)}\]
\[f=x^n-1,f'=(n-1)x^{n-2}\]
\[g'=\ln(x+\sqrt{x^2+a)}\]
not sure wat to use for g
Jonask
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sry \[f=x^{n-1}\]
hartnn
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let me try lil bit different,
\(I_n=\huge\int \frac{x^n}{\sqrt{x^2+a}}dx=\huge\int \frac{x.x^{n-1}}{\sqrt{x^2+a}}dx\)
now let f(x) = x^{n-1} , g'(x) =x/(...)
hartnn
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just a thought, donno whether it'll be useful
Jonask
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thanks it works very well,i tried it,just too lazy to post
hartnn
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oh, it works...good to hear :)