Advertisements
Advertisements
प्रश्न
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Advertisements
उत्तर
Let I = `int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Let 20 - 12ex = A(3ex - 4) + B `"d"/"dx"`(3ex - 4)
= 3 Aex - 4A + 3Bex
∴ 20 - 12ex = (3A + 3B)ex - 4A
Comparing the coefficients of ex and constant term on both sides, we get
- 4A = 20 and 3A + 3B = - 12
Solving these equations, we get
A = -5 and B = 1
∴ I = `int (-5(3"e"^"x" - 4) + 3"e"^"x")/(3"e"^"x" - 4)`dx
`= - 5 int "dx" + int (3"e"^"x")/(3"e"^"x" - 4)` dx
∴ I = - 5x + log `|(3"e"^"x" - 4)|` + c ....`[int ("f" '("x"))/("f" ("x")) "dx" = log |f ("x")| + "c"]`
Notes
The answer in the textbook is incorrect.
APPEARS IN
संबंधित प्रश्न
Evaluate :`intxlogxdx`
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of
Evaluate the following integrals : `int cos^2x.dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : `(4e^x - 25)/(2e^x - 5)`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Choose the correct options from the given alternatives :
`int f x^x (1 + log x)*dx`
`int logx/(log ex)^2*dx` = ______.
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate: `int "x" * "e"^"2x"` dx
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int ("e"^(3x))/("e"^(3x) + 1) "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)`
`int x^3"e"^(x^2) "d"x`
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
`int secx/(secx - tanx)dx` equals ______.
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int(1+x+x^2/(2!))dx`
