Advertisements
Advertisements
प्रश्न
`int x^2sqrt("a"^2 - x^6) "d"x`
Advertisements
उत्तर
Let I = `int x^2sqrt("a"^2 - x^6) "d"x`
= `int sqrt("a"^2 - (x^3)^2)*x^2 "d"x`
Put x3 = t
∴ 3x2dx = dt
∴ x2dx = `1/3 "dt"`
∴ I = `1/3 int sqrt("a"^2 - "t"^2) "dt"`
= `1/3["t"/2 sqrt("a"^2 - "t"^2) + "a"^2/2 sin^-1 ("t"/"a")] + "c"`
∴ I = `1/6[x^3 sqrt("a"^2 - x^6) + "a"^2 sin^-1 (x^3/"a")] + "c"`
APPEARS IN
संबंधित प्रश्न
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`1/(x(x^4 - 1))`
`int (xdx)/((x - 1)(x - 2))` equals:
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Evaluate: `int (5"x"^2 + 20"x" + 6)/("x"^3 + 2"x"^2 + "x")` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int sqrt(4^x(4^x + 4)) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int x^3tan^(-1)x "d"x`
`int x sin2x cos5x "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int ("d"x)/(x^3 - 1)`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
Evaluate: `int (dx)/(2 + cos x - sin x)`
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
When is a rational function \[\frac{P(x)}{Q(x)}\] called proper?
When is a rational function called improper?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
Which partial form is appropriate for \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})^{2}(x-\mathrm{b})}\]?
For \[\frac{5x-5}{(x-2)(x-3)}=\frac{\mathrm{A}}{x-2}+\frac{\mathrm{B}}{x-3}\], which equation results after clearing denominators?
