Advertisements
Advertisements
Question
`int x^2sqrt("a"^2 - x^6) "d"x`
Advertisements
Solution
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
RELATED QUESTIONS
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
`int (xdx)/((x - 1)(x - 2))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Find :
`∫ sin(x-a)/sin(x+a)dx`
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
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 with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `sec^2x sqrt(7 + 2 tan x - tan^2 x)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int (sinx)/(sin3x) "d"x`
`int sec^3x "d"x`
`int ("d"x)/(2 + 3tanx)`
`int ("d"x)/(x^3 - 1)`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int 1/(sinx(3 + 2cosx)) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c 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
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Which expression defines a rational function?
When is a rational function called improper?
A proper rational function can be expressed as a sum of simpler rational functions called what?
Which decomposition corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x^{2}+\mathrm{b}x+\mathrm{c})}\]?
What is the result of dividing \[x^{2}+1\] by \[x^{2}-5x+6\]?
Which decomposition is correct for \[\frac{x^{2}+1}{x^{2}-5x+6}\]?
Evaluate \[\int\frac{x^{2}+1}{x^{2}-5x+6}\,dx\].
What numerator is used for an irreducible quadratic factor?
