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evaluate the inetgral, int(4x+1)/(sqrt(x...

evaluate the inetgral, `int(4x+1)/(sqrt(x^(2)+3x+2))dx`

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`I=int(4x+1)/(sqrt(x^(2)+3x+2))dx`
Express, `4x+1=lambda{(d)/(dx)(x^(2)+3x+2)}+mu`
`:. 4x+1=lambda(2x+3)+mu`
On comparison,
`lambda=2`, `mu=-5`
`:.` Now, `l=2int(2x+3)/(sqrt(x^(2)+3x+2))dx-5int(1)/(sqrt(x^(2)+3x+2))dx`
Put `x^(2)+3x+2=t`
`x^(2)+3x+2=(x+(3)/(2))^(2)-((1)/(2))^(2)`
`:.(2x+3)dx=dt`
`impliesl=2int(dt)/(sqrt(t))-5int(1)/(sqrt((x+(3)/(2))^(2)-((1)/(2))^(2)))dx`
`l=2*2sqrt(t)-5log|x+(3)/(2)+sqrt(x^(2)+3x+2)|+C`
`:.l=4sqrt(x^(2)+3x+2)-5log|x+(3)/(2)+sqrt(x^(2)+3x+2)|+C`
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