Evaluate: `int(5x+3)/(sqrt(x^2+4x+10))\ dx`

Explanation

`" Let "I = int (5x+3)/(sqrt(x^(2)+4x+10))dx`
`" Let " 5x+3 =A (d)/(dx)(x^(2)+4x+10)+B`
` rArr 5x+3 =A (2x+4)+B`
Comparing coefficeints of x
`5=2A rArr A=(5)/(2)`
Comparing constant terms
`3=4A+B`
`rArr 3=10 +B`
`rArr B=-7`
`:. I= int ((5)/(2) (2x+4)-7)/(sqrt(x^(2)+4x+10))dx`
`=(5)/(2)int (2x+4)/(sqrt(x^(2) +4x+10))dx-7 int (1)/(sqrt(x^(2)+4x+10))dx`
`underset(rArr (2x+4)dx=dt)underset(" Let "x^(2)+4x+10=t)(" For first integral ")`
`=(5)/(2) int (1)/(sqrt(t))dt-7 int (1)/((x^(2)+4x+4)+6)dx`
`=(5)/(2) int t^(-1//2)dt-7 int (1)/(sqrt((x+2)^(2)+(sqrt(6))^(2)))dx`
`=(5)/(2) .(t^(1//2))/(1//2)`
`-7log |x+2+sqrt((x+2)^(2)+(sqrt(6))^(2))|+c`
`=5sqrt(x^(2)+4x+10)`
`-7 log | x+2+sqrt(x^(2)+4x+10)|+c`