Advertisements
Advertisements
प्रश्न
`int xcos^3x "d"x`
Advertisements
उत्तर
Let I = `int xcos^3x "d"x`
cos3x = 4cos3x − 3cosx
∴ 4cos3x = 3cos x + cos 3x
∴ cos3x = `1/4 (3cos x + cos 3x)`
∴ I = `1/4 int x (3cos x + cos 3x) "d"x`
= `1/4[x int (3cosx + cos3x) "d"x - int{"d"/("d"x)(x) int(3cos x + cos 3x)"d"x}"d"x]`
= `1/4[x(3sinx + (sin3x)/3) - int 1(3sinx + (sin3x)/3)"d"x]`
= `1/4[3x sinx + x/3 sin 3x - (-3 cosx - 1/3 * (cos3x)/3)] + "c"`
∴ I = `1/4(3x sinx + x/3 sin 3x + 3 cos x + 1/9 cos 3x) + "c"`
APPEARS IN
संबंधित प्रश्न
Find : `int x^2/(x^4+x^2-2) dx`
Find: `I=intdx/(sinx+sin2x)`
Evaluate: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`(x^3 + x + 1)/(x^2 -1)`
Integrate the rational function:
`1/(x(x^4 - 1))`
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
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 : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
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 w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int 1/(4x^2 - 20x + 17) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int ("d"x)/(2 + 3tanx)`
`int x^3tan^(-1)x "d"x`
`int x sin2x cos5x "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Choose the correct alternative:
`int ((x^3 + 3x^2 + 3x + 1))/(x + 1)^5 "d"x` =
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
Evaluate `int x log x "d"x`
`int x/((x - 1)^2 (x + 2)) "d"x`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
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.
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 (2x^2 - 3)/((x^2 - 5)(x^2 + 4))dx`
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
