Advertisements
Advertisements
प्रश्न
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
Advertisements
उत्तर
Let I = `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
Let `2"e"^"x" - 3 = "A" (4"e"^"x" + 1) + "B" "d"/"dx" (4"e"^"x" + 1)`
∴ `2"e"^"x" - 3 = (4"A" + 4"B")"e"^"x" + "A"`
Comparing the coefficients of `"e"^"x"` and constant term on both sides, we get
4A + 4B = 2 and A = - 3
Solving these equations, we get
B = `7/2`
∴ I = `(- 3 (4"e"^"x" + 1) + 7/2(4"e"^"x"))/(4"e"^"x" + 1)` dx
`= - 3 int "dx" + 7/2 int (4"e"^"x")/(4"e"^"x" + 1)`dx
∴ I = `- 3"x" + 7/2 log |4"e"^"x" + 1|` + c ...`[because int ("f" '("x"))/("f"("x")) "dx" = log |"f"("x")| + "c"]`
APPEARS IN
संबंधित प्रश्न
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Write a value of\[\int \log_e x\ dx\].
Evaluate the following integrals : tan2x dx
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Integrate the following functions w.r.t. x : `int (1)/(cosx - sinx).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c
Evaluate: `int log ("x"^2 + "x")` dx
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int 1/(xsin^2(logx)) "d"x`
`int (cos2x)/(sin^2x) "d"x`
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int ("d"x)/(sinx cosx + 2cos^2x)` = ______.
`int (f^'(x))/(f(x))dx` = ______ + c.
`int (logx)^2/x dx` = ______.
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate `int(1 + x + x^2/(2!))dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
