Advertisements
Advertisements
प्रश्न
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
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: `∫8/((x+2)(x^2+4))dx`
Integrate the rational function:
`x/((x^2+1)(x - 1))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
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: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Evaluate: `int "3x - 2"/(("x + 1")^2("x + 3"))` dx
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int x^7/(1 + x^4)^2 "d"x`
`int sqrt((9 + x)/(9 - x)) "d"x`
`int sec^3x "d"x`
`int sin(logx) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
Evaluate:
`int (5e^x)/((e^x + 1)(e^(2x) + 9)) dx`
`int (3"e"^(2x) + 5)/(4"e"^(2x) - 5) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
`int (5(x^6 + 1))/(x^2 + 1) "d"x` = x5 – ______ x3 + 5x + c
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
`int 1/(x^2 + 1)^2 dx` = ______.
