Advertisements
Advertisements
Question
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Advertisements
Solution
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
RELATED QUESTIONS
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
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:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`((x^2 +1)(x^2 + 2))/((x^2 + 3)(x^2+ 4))`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the rational function:
`1/(x(x^4 - 1))`
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
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 w.r.t.x : `(1)/(sinx + sin2x)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
`int x sin2x cos5x "d"x`
`int ("d"x)/(x^3 - 1)`
`int xcos^3x "d"x`
`int (sin2x)/(3sin^4x - 4sin^2x + 1) "d"x`
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
`int 1/(x^2 + 1)^2 dx` = ______.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
