Advertisements
Advertisements
Question
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Advertisements
Solution
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
RELATED QUESTIONS
Integrate the functions:
`1/(x-sqrtx)`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int (log x)/(log ex)^2` dx = _________
`int sqrt(1 + sin2x) dx`
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"`
Evaluate `int(3x^2 - 5)^2 "d"x`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int(5x^2-6x+3)/(2x-3) dx`
If f '(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
