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Evaluate: (i) intx/(x^4+x^2+1)\ dx (ii) ...

Evaluate: (i) `intx/(x^4+x^2+1)\ dx` (ii) `int(e^x)/(e^(2x)+6e^x+5)\ dx`

Text Solution

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(i) Consider, `I=frac{x}{x^{4}+x^{2}+1} d x`
put `x^{2}+1=t Rightarrow x^{2}=t-1 Rightarrow x^{4}=(t-1)^{2}`
`2 xdx=dt`
`xdx=frac{1}{2} dt`
`Rightarrow I=int frac{1}{2[(t-1)^{2}+t]} d t`
`=frac{1}{2} int frac{1}{t^{2}+1-2 t+t} d t`
...
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