Advertisements
Advertisements
Question
`int sqrt(x^2 - 8x + 7) dx` is equal to:-
Options
`1/2(x - 4) sqrt(x^2 - 8x + 7) + 9 log|x - 4 + sqrt(x^2 - 8x + 7)| + c`
`1/2(x - 4) sqrt(x^2 - 8x + 7) + 9 log|x + 4 + sqrt(x^2 - 8x + 7)| + c`
`1/2(x - 4) sqrt(x^2 - 8x + 7) + 3sqrt(2) log|x - 4 + sqrt(x^2 - 8x + 7)| + c`
`1/2(x - 4) sqrt(x^2 - 8x + 7) - 9/2 log|x - 4 + sqrt(x^2 - 8x + 7)| + c`
Solution
`1/2(x - 4) sqrt(x^2 - 8x + 7) - 9/2 log|x - 4 + sqrt(x^2 - 8x + 7)| + c`
Explanation:
`int sqrt(x^2 - 8x + 7) dx`
= `int sqrt(x^2 - 8x + 16 + 7) dx`
= `int sqrt((x - 4^2 - 9)) dx`
We have, `int sqrt(x^2 - a^2) dx`
= `x/2 sqrt(x^2 - a^2) - a^2/2 log|x + sqrt(x^2 - a^2)| + c`
Replace `x` by `x - 4, a^2 = 9, a = 3`
∴ `int sqrt((x - 4)^2 - 9x) dx`
= `((x - 4))/2 sqrt(x^2 - 8x + 7) - 9/2 log |(x - 4) + sqrt(x^2 - 8x + 7)| + c`