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Find the following integrals (i) int(1...

Find the following integrals
`(i) int(1)/(x^(2)+2x+2)dx` `(ii) int(1)/(3x^(2)+13x-10)dx`

Text Solution

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`(i) l=int(1)/(x^(2)+2x+2)dx=int(1)/((x^(2)+2x+1)+1)dx`
`:.l=int(1)/((x+1)^(2)+1)dx=tan^(-1)(x+1)+C`
`(ii) l=int(1)/(3x^(2)+13x-10)dx=(1)/(3)int(1)/(x^(2)+(13)/(3)x-(10)/(3))dx`
`:.l=(1)/(3)int(1)/((x+(13)/(6))^(2)-((17)/(6))^(2))dx`, put `x+(13)/(6)=t`
`impliesdx=dt`
`:.l=(1)/(3)int(1)/(t^(2)-((17)/(6))^(2))dt=(1)/(2*3*(17)/(6))log|(t-(17)/(6))/(t+(17)/(6))|+C_(1)`
`:.l=(1)/(17)log|(x+(13)/(6)-(17)/(6))/(x+(13)/(6)+(17)/(6))|+C`
`:.l=(1)/(17)log|(6x-4)/(6x+30)|+C`
`:.l=(1)/(17)log|(3x-2)/(x+5)|+C`
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