Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Advertisements
उत्तर
Let I = `int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Let 2ex + 5 = A(2ex + 1) + B `"d"/"dx"`(2ex + 1)
= 2 Aex + 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)`dx
`= 5 int "dx" - 4 int (2"e"^"x")/(2"e"^"x" + 1)`dx
∴ I = 5x - 4 log `|2"e"^"x" + 1|` + c ....`[int ("f" '("x"))/("f" ("x")) "dx" = log |f ("x")| + "c"]`
Notes
The answer in the textbook is incorrect.
APPEARS IN
संबंधित प्रश्न
Evaluate : `∫1/(cos^4x+sin^4x)dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`1/(1 + cot x)`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals : `intsqrt(1 + sin 5x).dx`
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int 1/("x" log "x")`dx
`int sqrt(1 + "x"^2) "dx"` =
`int (7x + 9)^13 "d"x` ______ + c
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
`int secx/(secx - tanx)dx` equals ______.
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
