Advertisements
Advertisements
Question
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Advertisements
Solution
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + 2
Explanation:
f(x) = ∫f '(x) dx
`= int (1/"x" + "x")` dx
f(x) = log |x| + `"x"^2/2 + "c"` ...(i)
f(1) = `5/2`
f(1) = log 1 + `1^2/2` + c
∴ `5/2 = 0 + 1/2 + "c"` ...(∵ log 1 = 0)
∴ c = `5/2 - 1/2`
∴ = `4/2` = 2
∴ c = 2
∴ f(x) = log |x| + `"x"^2/2` + 2
Notes
The answer in the textbook is incorrect.
RELATED QUESTIONS
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals : `int cos^2x.dx`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
`int logx/(log ex)^2*dx` = ______.
Evaluate the following.
`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
Evaluate the following.
`int(20 - 12"e"^"x")/(3"e"^"x" - 4) "dx"`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If \[x=g(t)\] and \[dx=g'(t)dt\], which formula represents integration by substitution?
After putting \[t=\cos x\], which integral is obtained from \[\int\sin^2x\cos^2x(\sin x)\,dx\]?
For indefinite integrals, what should be done after integration in the new variable?
