Advertisements
Advertisements
प्रश्न
`int x sin2x cos5x "d"x`
उत्तर
Let I = `int x sin2x cos5x "d"x`
= `1/2 int x (2 sin 2x cos 5x) "d"x`
= `1/2 int x [sin (2x + 5x) + sin(2x - 5x)] "d"x`
= `1/2 int x [sin 7x - sin (-3x)] "d"x`
= `1/2 int x (sin 7x - sin 3x) "d"x`
= `1/2 int x sin 7x "d"x - 1/2 int x sin 3x "d"x`
= `1/2 [x int sin 7x "d"x - int {"d"/("d"x) (x) int sin 7x "d"x}"d"x] - 1/2 [x int sin 3x "d"x - int {"d"/("d"x)(x) int sin 3x "d"x}"d"x]`
= `1/2[x(- (cos 7x)/7) - int 1* ((-cos 7x)/7) "d"x] - 1/2[x((-cos 3x)/3) - int 1* ((-cos 3x)/3) "d"x]`
= `1/2 ((-x cos 7x)/7 + 1/7 int cos 7x "d"x) - 1/2((-x cos 3x)/3 + 1/3 int cos 3x "d"x)`
= `1/2[(-x cos 7x)/7 + 1/7((sin7x)/7)] - 1/2[(-x cos 3x)/3 + 1/3((sin 3x)/3)] + "c"`
∴ I = `1/98 sin 7x - 1/14 x cos 7x - 1/18 sin 3x + 1/6 x cos 3x + "c"`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Find : `int x^2/(x^4+x^2-2) dx`
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
Integrate the rational function:
`1/(x^4 - 1)`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the rational function:
`1/(x(x^4 - 1))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
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 : `(12x + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`
Integrate the following w.r.t. x : `(1)/(x^3 - 1)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x : `(x + 5)/(x^3 + 3x^2 - x - 3)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int x^7/(1 + x^4)^2 "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int (sinx)/(sin3x) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int ("d"x)/(x^3 - 1)`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
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 x/((x - 1)^2 (x + 2)) "d"x`
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int sqrt(tanx) "d"x` (Hint: Put tanx = t2)
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 f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
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 ______.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate: `int (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
