Advertisements
Advertisements
प्रश्न
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
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 :`intxlogxdx`
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`1/(1 - tan x)`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals : `int cos^2x.dx`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t. x:
`(10x^9 10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : cos7x
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Integrate the following functions w.r.t. x : `(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int (1)/(5 - 4x - 3x^2).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sqrt(3)sinx).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following : `int (logx)2.dx`
`int logx/(log ex)^2*dx` = ______.
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int "x" * "e"^"2x"` dx
Evaluate: `int log ("x"^2 + "x")` dx
Evaluate: `int "e"^sqrt"x"` dx
`int 1/sqrt((x - 3)(x + 2))` dx = ______.
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int sqrt(1 + sin2x) "d"x`
`int 1/(xsin^2(logx)) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
`int(5x + 2)/(3x - 4) dx` = ______
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
The value of `sqrt(2) int (sinx dx)/(sin(x - π/4))` is ______.
`int cos^3x dx` = ______.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate `int(1+ x + x^2/(2!)) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate:
`intsqrt(sec x/2 - 1)dx`
