Advertisements
Advertisements
प्रश्न
Evaluate the following : `int x^2.log x.dx`
Advertisements
उत्तर
Let I = `int x^2.logx.dx`
= `int log x.x^2.dx`
= `(logx) int x^2.dx - int[{d/dx (logx) int x^2.dx}].dx`
= `(logx).x^3/(3) - int (1)/x.x^3/(3).dx`
= `x^3/(3) logx - (1)/(3) int x^2.dx`
= `x^3/(3) logx - (1)/(3)(x^3/3) + c`
= `x^3/(9)(3.logx - 1) + c`.
APPEARS IN
संबंधित प्रश्न
`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 `(xe^x)/(1+x)^2`.
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in e2x sin x.
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int x^3 e^(x^2)`dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int 1/sqrt(2x^2 - 5) "d"x`
`int sin4x cos3x "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/x "d"x` = ______ + c
Evaluate `int 1/(x log x) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
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`
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.
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
`int_0^1 x tan^-1 x dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^2e^(4x)dx`
The value of `inta^x.e^x dx` equals
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
`∫ sin^(−1)` xdx is equal to ______.
