Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Advertisements
उत्तर
Let I = `int (1 + x)/(x.sin (x + log x)).dx`
= `int (1)/(sin(x + logx)).((1 + x)/x).dx`
= `int (1)/(sin(x + log x)).(1/x + 1).dx`
Put x + log x = t
∴ `(1 + 1/x).dx` = dt
∴ I = `int (1)/sint dt = int "cosec" t dt`
= log |cosec t – cot t| + c
= log |cosec(x + log x) – cot(x + logx)|+ c.
संबंधित प्रश्न
Evaluate :`intxlogxdx`
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`x^2/(2+ 3x^3)^3`
Integrate the functions:
sec2(7 – 4x)
`int (dx)/(sin^2 x cos^2 x)` equals:
Write a value of
Write a value of
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Evaluate the following integrals: `int (2x - 7)/sqrt(4x - 1).dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Evaluate the following:
`int sinx/(sin 3x) dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
If f'(x) = x2 + 5 and f(0) = −1, then find the value of f(x).
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Choose the correct alternative from the following.
`int "dx"/(("x" - "x"^2))`=
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int sec^6 x tan x "d"x` = ______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Which standard substitution is used for \[\sqrt{\mathrm{a}^2-x^2}\], \[\frac{1}{\sqrt{\mathrm{a}^2-x^2}}\], or \[\mathrm{a}^2-x^2\]?
Which standard substitution is used for \[\sqrt{x^2+\mathrm{a}^2}\], \[\frac{1}{\sqrt{x^2+\mathrm{a}^2}}\], or \[x^2+\mathrm{a}^2\]?
Which substitution is appropriate for \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}}\] or \[\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\]?
