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 log 2x.
Integrate the function in x sin−1 x.
Integrate the function in x cos-1 x.
Integrate the function in x sec2 x.
Integrate the function in tan-1 x.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
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 with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (sinx)/(1 + sin x) "d"x`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Evaluate `int 1/(x log x) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
`int(logx)^2dx` equals ______.
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
`int_0^1 x tan^-1 x dx` = ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(1-x)^-2 dx` = ______
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate `int tan^-1x dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x 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)
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^3 e^(x^2)dx`
