Advertisements
Advertisements
प्रश्न
`int ("d"x)/(2 + 3tanx)`
उत्तर
Let I = `int 1/(2 + 3tanx) "d"x`
= `int 1/(2 + 3(sinx/cosx)) "d"x`
= `int cosx/(2cosx + 3sinx) "d"x`
Let cos x = `"A"(2cosx + 3 sinx) + "B""d"/("d"x) (2cosx + 3sinx)`
= A(2cos x + 3sin x) + B(−2sin x + 3cos x)
∴ cos x + 0⋅sinx = cosx (2A + 3B) + sinx (3A − 2B)
By equating the coefficients on both sides, we get
2A + 3B = 1 and 3A − 2B = 0
Solving these equations, we get
A = `2/13` and B = `3/13`
∴ cos x = `2/13 (2 cos x + 3 sin x) + 3/13 (-2 sin x + 3 cos x)`
∴ I = `int (2/13(2cos x + 3sin x) + 3/13(-2 sinx + 3cos x))/(2cosx + 3sin x) "d"x`
∴ I = `2/13 int "d"x + 3/13 int (-2sinx + 3cosx)/(2cosx + 3sinx) "d"x`
∴ I = `2/13x + 3/13 log |2cos + 3sinx| + "c"` ........`[∵ int ("f'"(x))/("f"(x)) "d"x = log |"f"(x)| + "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:
`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:
`(3x + 5)/(x^3 - x^2 - x + 1)`
Integrate the rational function:
`(2x - 3)/((x^2 -1)(2x + 3))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the rational function:
`(3x -1)/(x + 2)^2`
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:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
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]
`int (xdx)/((x - 1)(x - 2))` equals:
`int (dx)/(x(x^2 + 1))` equals:
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
Find `int (2cos x)/((1-sinx)(1+sin^2 x)) dx`
Find :
`∫ sin(x-a)/sin(x+a)dx`
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 + 3)/(6x^2 + 13x - 63)`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
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(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 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: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
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 1/(x(x^3 - 1)) "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int 1/(sinx(3 + 2cosx)) "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 = ?
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
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 x^2"e"^(4x) "d"x`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
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.
Evaluate: `int (dx)/(2 + cos x - sin x)`
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If `int 1/((x^2 + 4)(x^2 + 9))dx = A tan^-1 x/2 + B tan^-1(x/3) + C`, then A – B = ______.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
