Advertisements
Advertisements
प्रश्न
Evaluate `int x log x "d"x`
Advertisements
उत्तर
Let I = `int x* log x "d"x`
= `log x int x"d"x - int["d"/("d"x) (log x) int x"d"x] "d"x`
= `log x* x^2/2 - int[1/x xx x^2/2] "d"x`
= `x^2/2 log x - 1/2 int x "d"x`
= `x^2/2 log x - 1/2* x^2/2 + "c"`
∴ I = `x^2/2 log x - x^2/4 + "c"`
APPEARS IN
संबंधित प्रश्न
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int x^3tan^(-1)x "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Let g : (0, ∞) `rightarrow` R be a differentiable function such that `int((x(cosx - sinx))/(e^x + 1) + (g(x)(e^x + 1 - xe^x))/(e^x + 1)^2)dx = (xg(x))/(e^x + 1) + c`, for all x > 0, where c is an arbitrary constant. Then ______.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
