Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Advertisements
उत्तर
Let I = `int (x + 1)sqrt(2x^2 + 3)`
Let x + 1 = `"A"[d/dx (2x^2 + 3)] + "B"`
= A (4x) + B
= 4Ax + B
Comparing the coefficients of and constant on both sides, we get
4A = 1, B = 1
∴ A = `(1)/(4), "B"` = 1
∴ x + 1 = `(1)/(4)(4x) + 1`
∴ I = `int [1/4 (4x) + 1]sqrt(2x^2 + 3).dx`
= `(1)/(4) int 4x sqrt(2x^2 + 3).dx + int sqrt(2x^2 + 3).dx`.
= I1 + I2
In I1 = put 2x2 + 3 = t
∴ 4x.dx = dt
∴ I1 = `(1)/(4) int t^(12).dt`
= `(1)/(4)(t^(3/2)/(3/2)) + c_1`
= `(1)/(6)(2x^2 + 3)^(3/2) + c_1`
I2 = `int sqrt(2x^2 + 3).dx`
= `sqrt(2) int sqrt(x^2 + 3/2).dx`
= `sqrt(2)[x/2sqrt(x^2 + 3/2) + ((3/2))/(2)log|x + sqrt(x^2 + 3/2)|] + c_2`
= `sqrt(2)[x/2sqrt(x^2 + 3/2) + (3)/(4)log|x + sqrt(x^2 + 3/2)|] + c_2`
∴ I = `(1)/(6)(2x^2 + 3)^(3/2) + sqrt(2)[x/2 sqrt(x^2 + 3/2) + (3)/(4) log|x + sqrt(x^2 + 3/2)|] + c`, where c = c1 + c2.
APPEARS IN
संबंधित प्रश्न
Integrate the function in x sec2 x.
Integrate the function in x (log x)2.
Integrate the function in `(xe^x)/(1+x)^2`.
`int e^x sec x (1 + tan x) dx` equals:
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following: `int logx/x.dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : sec4x cosec2x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
`int (sinx)/(1 + sin x) "d"x`
`int sin4x cos3x "d"x`
`int 1/x "d"x` = ______ + c
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Find `int_0^1 x(tan^-1x) "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
Solve: `int sqrt(4x^2 + 5)dx`
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
`int logx dx = x(1+logx)+c`
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
Evaluate:
`int1/(x^2 + 25)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
The value of `inta^x.e^x dx` equals
`∫ sin^(−1)` xdx is equal to ______.
Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]
