Advertisements
Advertisements
प्रश्न
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`
Advertisements
उत्तर
Let `intv dx`= w ......(i)
Then, `(dw)/(dx)` = v ......(ii)
Now, `d/(dx)(u, w) = u. d/(dx)(w) + w d/(dx)(u)`
= `u.v + w (du)/(dx)` ......[From (ii)]
By definition of integration
u.w = `int[u.v + w . (du)/(dx)]dx`
∴ u.w = `intu.v dx + intw. (du)/(dx) dx`
∴ `intu.v dx = u.w - intw. (du)/(dx) dx`
= `uintv dx - int[(du)/(dx) intv. dx]dx` ......[Using (i)]
Hence, `intlogx dx = xlogx - int 1/x x xx dx`
= x log x – x + C
APPEARS IN
संबंधित प्रश्न
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in x log x.
Integrate the function in x log 2x.
Integrate the function in x cos-1 x.
Integrate the function in x sec2 x.
Integrate the function in `(xe^x)/(1+x)^2`.
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int x^2 e^4x`dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int 1/(4x + 5x^(-11)) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int sin4x cos3x "d"x`
Evaluate `int 1/(x(x - 1)) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int log x * [log ("e"x)]^-2` dx = ?
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
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 ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`int1/sqrt(x^2 - a^2) dx` = ______
`intsqrt(1+x) dx` = ______
`int1/(x+sqrt(x)) dx` = ______
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int e^(ax)*cos(bx + c)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 the following:
`intx^3e^(x^2)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
The value of `inta^x.e^x dx` equals
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
