Advertisements
Advertisements
प्रश्न
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Advertisements
उत्तर
`I=∫(x+2)/sqrt(x^2+5x+6)dx `
Multiplying and dividing by 2, we get
`I=1/2∫(2x+4)/sqrt(x^2+5x+6)dx `
Adding and subtracting 1 to the numerator, we get:
`I=1/2∫(2x+4+1-1)/sqrt(x^2+5x+6)dx`
` I=1/2∫(2x+5)/sqrt(x2+5x+6)dx -1/2∫1/sqrt(x^2+5x+6)dx`
`"Let" I_1=1/2∫(2x+5)/sqrt(x^2+5x+6)dx `
Put x2+5x+6=t
Differentiating with respect to x, we get:
(2x+5)dx=dt
`I_1=intdt/sqrtt`
`I_1=2sqrtt+c`
`I_1=2sqrt(x^2+5x+6)+c`
`1/2 int "dt"/sqrt t =∫1/sqrt(x^2+5x+(5/2)^2-(5/2)^2+6)dx`
`1/2 int "dt"/sqrt t - 1/2 int "dx"/sqrt(x^2+5x+6 + (5/2)^2 - 25/4)dx`
`1/2 "t"^(1/2)/(1/2) int 1/sqrt((x+5/2)^2-(1/2)^2)dx`
`= 1/2 xx 2 xx "t"^(1/2) - 1/2 |"log" x + 5/2 + sqrt (x^2 + 5x + 6)| + "C"`
`= sqrt (x^2 + 5x + 6) - 1/2 "log" |sqrt (x^2 + 5x + 6)| + "C"`
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`(1+ log x)^2/x`
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
`int ("e"^x(x - 1))/(x^2) "d"x` = ______
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int 1/(xsin^2(logx)) "d"x`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int "e"^(sin^-1 x) ((x + sqrt(1 - x^2))/(sqrt1 - x^2)) "dx" = ?`
`int ("d"x)/(x(x^4 + 1))` = ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
Evaluate `int1/(x(x - 1))dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
