Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Advertisements
उत्तर
Let I = `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Let 3x + 4 = `"A"[d/dx (2x^2 + 2x + 1)\ + "B"` ...(i)
3x + 4 = A(4x + 2) + B
∴ 3x + 4 = (4A)x + (2A + B)
Consider,
4A = 3 and 2A + B = 4
∴ A = `(3)/(4) and 2(3/4) + "B"` = 4
∴ B = `4- 3/2`
∴ B = `8 - 3/2`
∴ B = `(5)/(2)`
From (i),
(3x + 4) = `3/4 d/dx (2x^2 + 2x + 1) + 5/2` ...(ii)
The required integral is,
I = `int ((3/4.d/dx (2x^2 + 2x + 1) + 5/2)/(sqrt(2x^2 + 2x + 1))dx`
I = `3/4 int (d/dx (2x^2 + 2x + 1))/(sqrt(2x^2 + 2x + 1)) dx + 5/2 int 1/ (sqrt(2x^2 + 2x + 1))dx`
I = `3/4 . 2 . sqrt(2x^2 + 2x + 1) + 5/2 . 1/sqrt2 int 1/sqrt(x^2 + x + 1/2)dx + c_1` ...`int(f'(x))/sqrtf(x)dx = 2 sqrtf(x) + c`
I = `3/2 sqrt(2x^2 + 2x + 1) + 5/(2sqrt2) int 1/sqrt((x^2 + x + 1/4) + 1/2 - 1/4)dx + c_1`
I = `3/2 sqrt(2x^2 + 2x + 1) + 5/(2sqrt2) int 1/ sqrt((x + 1/2)^2 + (1/2)^2)dx + c_1`
I = `3/2 sqrt(2x^2 + 2x + 1) + 5/(2sqrt2) log |(x + 1/2) + sqrt((x + 1/2)^2 + (1/2)^2)| + c_1 + c_2`
I = `3/2 sqrt(2x^2 + 2x + 1) + 5/(2sqrt2) log |(x + 1/2) + sqrt(x^2 + x + 1/2)| + c`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`e^(2x+3)`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Evaluate: `int 1/(x(x-1)) dx`
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Choose the correct options from the given alternatives :
`int (cos2x - 1)/(cos2x + 1)*dx` =
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
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 (sin4x)/(cos 2x) "d"x`
`int (2 + cot x - "cosec"^2x) "e"^x "d"x`
`int(log(logx))/x "d"x`
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int(5x + 2)/(3x - 4) dx` = ______
If `int sinx/(sin^3x + cos^3x)dx = α log_e |1 + tan x| + β log_e |1 - tan x + tan^2x| + γ tan^-1 ((2tanx - 1)/sqrt(3)) + C`, when C is constant of integration, then the value of 18(α + β + γ2) is ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate `int1/(x(x-1))dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
Evaluate:
`intsqrt(sec x/2 - 1)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
