Advertisements
Advertisements
Question
For `int ("x - 1")/("x + 1")^3 "e"^"x" "dx" = "e"^"x"` f(x) + c, f(x) = (x + 1)2.
Options
True
False
Advertisements
Solution
This statement is false.
Explanation:
Let I =`(("x" - 1))/(("x" + 1)^3) * "e"^"x"` dx
`= int "e"^"x" [(("x" + 1) - 2)/("x"+ 1)^3]` dx
`= int "e"^"x" [1/("x" + 1)^2 - 2/("x" + 1)^3]` dx
`= int "e"^"x" [("x" + 1)^-2 - 2("x" + 1)^-3]` dx
Put f(x) = (x + 1)-2
∴ f '(x) = − 2 (x + 1)−3
∴ I = `"e"^"x" ["f"("x") + "f" '("x")]` dx
`= "e"^"x" * "f"("x")` + c
`= "e"^"x" * ("x + 1")^-2` + c
∴ f(x) = (x + 1)−2
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`1/(x(x^n + 1))` [Hint: multiply numerator and denominator by xn − 1 and put xn = t]
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`1/(x(x^4 - 1))`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Evaluate: `int 1/("x"("x"^5 + 1))` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int sqrt((9 + x)/(9 - x)) "d"x`
`int sin(logx) "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
`int 1/x^3 [log x^x]^2 "d"x` = p(log x)3 + c Then p = ______
`int 1/(4x^2 - 20x + 17) "d"x`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
If `int dx/sqrt(16 - 9x^2)` = A sin–1 (Bx) + C then A + B = ______.
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
If \[\int\frac{2x+3}{(x-1)(x^{2}+1)}\mathrm{d}x\] = \[=\log_{e}\left\{(x-1)^{\frac{5}{2}}\left(x^{2}+1\right)^{a}\right\}-\frac{1}{2}\tan^{-1}x+\mathrm{A}\] where A is an arbitrary constant, then the value of a is
What is done after long division, before writing the appropriate partial-fraction decomposition?
Which factorisation is correct for \[x^{2}-5x+6\]?
What numerator is used for an irreducible quadratic factor?
