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
Evaluate:
`int x^2/(x^4+x^2-2)dx`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`1/(x(x^4 - 1))`
Find `int(e^x dx)/((e^x - 1)^2 (e^x + 2))`
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^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 w.r.t. x : `(1)/(2sinx + sin2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
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: `(x + 5)/(x^3 + 3x^2 - x - 3)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int 1/(2 + cosx - sinx) "d"x`
`int "e"^x ((1 + x^2))/(1 + x)^2 "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int (x + sinx)/(1 - cosx) "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
Evaluate: `int (dx)/(2 + cos x - sin x)`
Evaluate`int(5x^2-6x+3)/(2x-3)dx`
