Advertisements
Advertisements
प्रश्न
Integrate the following with respect to x:
`(3x + 1)/(2x^2 - 2x + 3)`
Advertisements
उत्तर
Let 3x + 1 = `"A" "d"/("d"x) (2x^2 - 2x + 3) + "B"`
3x + 1 = A(4x – 2) + B
3x + 1 = 4Ax – 2A + B
4A = 3
⇒ A = `3/4`
– 2A + B = 1
`- 2 xx 3/4 + "B"` = 1
`3/2 + "B"` = 1
B = `1 + 3/2 = 5/2`
B = `5/2`
3x + 1 = `3/4 (4x - 2) + 5/2`
`int (3x + 1)/(2x^2 - 2x + 3) "d"x = int (3/4 (4x - 2) + 5/2)/(2x^2 - 2x + 3) "d"x`
= `3/4 int (4x - 2)/(2x^2 - 2x + 3) + 5/2 int ("d"x)/(2x^2 - 2x + 3)`
Put `2x^2 - 2x + 3` = t
`(4x - 2) "d"x` = dt
= `3/4 int "dt"/"t" + 5/2 int ("d"x)/(2(x^2 - x + 3/2))`
= `3/4 log |"t"| + 5/4 int ("d"x)/((x - 1/2)^2 - (1/2)^2 + 3/2)`
= `3/4 log |2x^2 - 2x + 3| + 5/4 int ("d"x)/((x - 1/2)^2 + - 1/4 + 3/2)`
= `3/4 log |2x^2 - 2x + 3| + 5/4 int ("d"x)/((x - 1/2)^2 + (6 - 1)/4`
= `3/4 log |2x^2 - 2x + 3| + 5/4 int ("d"x)/((x - 1/2)^2 + 5/4)`
= `3/4 log |2x^2 - 2x + 3| + 5/4 int ("d"x)/((x - 1/2)^2 + (sqrt(5)/2)^2`
`int ("d"x)/(x^2 + "a"^2) = 1/"a" tan^-1 (x/"a") + "c"`
= `3/4 log |2x^2 - 2x + 3| + 5/4 xx tan^-1 ((x - 1/2)/(sqrt(5)/2)) + "c"`
= `3/4 log |2x^2 - 2x + 3| + 5/4 xx 1/(sqrt(5)/2) tan^-1 ((2x - 1)/sqrt(5)) + "c"`
= `3/4 log |2x^2 - 2x + 3| + 5/4 xx 2/sqrt(5) tan^-1 ((2x - 1)/sqrt(5)) + "c"`
= `3/4 log |2x^2 - 2x + 3| + sqrt(5)/2 tan^-1 ((2x - 1)/sqrt(5)) + "c"`
APPEARS IN
संबंधित प्रश्न
Evaluate:`int(tansqrtx)/sqrtxdx`
Integrate the following functions with respect to x :
`[sqrt(x) + 1/sqrt(x)]^2`
Integrate the following functions with respect to x :
cos 3x cos 2x
Integrate the following with respect to x :
`x/sqrt(1 + x^2)`
Integrate the following with respect to x :
`(10x^9 + 10^x log_"e" 10)/(10^x + x^10)`
Integrate the following with respect to x :
x(1 – x)17
Integrate the following with respect to x :
sin5x cos3x
Integrate the following with respect to x :
`cosx/(cos(x - "a"))`
Integrate the following with respect to x:
x2 cos x
Integrate the following with respect to x:
`"e"^x ((x - 1)/(2x^2))`
Integrate the following with respect to x:
`"e"^x sec x(1 + tan x)`
Find the integrals of the following:
`1/sqrt(xx^2 + 4x + 2)`
Find the integrals of the following:
`1/sqrt(x^2 - 4x + 5)`
Integrate the following with respect to x:
`(2x + 1)/sqrt(9 + 4x - x^2)`
Integrate the following functions with respect to x:
`sqrt((6 - x)(x - 4))`
Integrate the following functions with respect to x:
`sqrt((x + 1)^2 - 4)`
Choose the correct alternative:
`int ("e"^x (1 + x))/(cos^2(x"e"^x)) "d"x` is
Choose the correct alternative:
`int tan^-1 sqrt((1 - cos 2x)/(1 + cos 2x)) "d"x` is
Choose the correct alternative:
`int sqrt((1 - x)/(1 + x)) "d"x` is
Choose the correct alternative:
`int (sec^2x)/(tan^2 x - 1) "d"x`
