Advertisements
Advertisements
Question
Evaluate the following integral:
`sqrt(9x^2 + 12x + 3) "d"x`
Solution
`sqrt(9x^2 + 12x + 3) "d"x = int sqrt(9(x^2 + 12/9x + 3/9)) "d"x`
= `3int sqrt(x^2 + 4/3x + 1/3 d"x`
= `3sqrt((x + 4/6)^2 - 16/36 + 1/3) "d"x`
= `3int sqrt((x + 2/3)^2 - 1/9) "d"x`
= `3 int sqrt((x + 2/3)^2 - (1/3)^2) "d"x`
= `3[((x + 2/3))/2 sqrt((x + 2/3)^2 - 1/9) - 1/((9)(2)) log |(x + 2/3) + sqrt((x + 2/3)^2 - 1/9)|] + "c"`
= `3[(3x + 2)/6 sqrt((3x + 2)^2/9 - 1/9) - 1/18 log|((3x + 2))/2 + sqrt((3x + 2/9)^2 - 1/9)|] + "c"`
= `3[((3x + 2))/6 sqrt(9x^2 + 12x + 3)/3 - 1/18 log |((3x + 2))/3 + sqrt(9x^2 + 12x + 3)/3|] + "c"`
= `((3x - 2))/6 sqrt(9x^2 + 12x + 3) - 1/6 log |(3x + 2) + sqrt(9x^2 + 12x + 3)| + "k"`
APPEARS IN
RELATED QUESTIONS
Integrate the following with respect to x.
`(3x^2 - 2x + 5)/((x - 1)(x^2 + 5))`
Integrate the following with respect to x.
2 cos x – 3 sin x + 4 sec2x – 5 cosec2x
Integrate the following with respect to x.
`1/(x^2(x^2 + 1))`
Integrate the following with respect to x.
`"e"^x [(x - 1)/(x + 1)^3]`
Integrate the following with respect to x.
`1/(2x^2 + 6x - 8)`
Integrate the following with respect to x.
`1/sqrt(9x^2 - 7)`
Choose the correct alternative:
`int 1/x^3 "d"x` is
Choose the correct alternative:
`int ("d"x)/sqrt(x^2 - 36) + "c"`
Choose the correct alternative:
`int_2^4 ("d"x)/x` is
Choose the correct alternative:
`int_0^1 sqrt(x^4 (1 - x)^2) "d"x` is
