Advertisements
Advertisements
प्रश्न
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Advertisements
उत्तर
Let I = `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Let 2ex + 5 = `"A" (2"e"^x + 1) + "B" "d"/("d"x) (2"e"^x + 1)`
= 2Aex + A + B(2ex)
∴ 2ex + 5 = (2A + 2B)ex + A
Comparing the coefficients of ex and constant term on both sides,
we get 2A + 2B = 2 and A = 5
Solving these equations, we get
B = – 4
∴ I = `int(5(2"e"^x + 1) - 4(2"e"^x))/(2"e"^x + 1) "d"x`
= `5int "d"x - 4int (2"e"^x)/(2"e"^x + 1) "d"x`
∴ I = 5x – 4log|2e + 1| + c ......`[because int ("f'"(x))/("f"(x)) "d"x = log|"f"(x)| + "c"]`
APPEARS IN
संबंधित प्रश्न
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`(3x - 1)/((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:
`1/(x^4 - 1)`
`int (dx)/(x(x^2 + 1))` equals:
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 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)`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Evaluate: `int ("3x" - 1)/("2x"^2 - "x" - 1)` dx
`int 1/(2 + cosx - sinx) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int ("d"x)/(x^3 - 1)`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
`int ((2logx + 3))/(x(3logx + 2)[(logx)^2 + 1]) "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 x^2"e"^(4x) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
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 ______.
Find: `int x^4/((x - 1)(x^2 + 1))dx`.
Find : `int (2x^2 + 3)/(x^2(x^2 + 9))dx; x ≠ 0`.
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
Evaluate.
`int (5x^2 - 6x + 3) / (2x -3) dx`
Evaluate:
`int (x + 7)/(x^2 + 4x + 7)dx`
