Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
उत्तर
Let I = `int (x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x).dx`
= `int a^(x + tan^-1x).((x^2 + 2)/(x^2 + 1)).dx`
Put x + tan–1x = t
∴ `(1 + 1/(1 + x^2)).dx` = dt
∴ `((1 + x^2 + 1)/(1 + x^2)).dx` = dt
∴ `((x^2 + 2)/(x^2 + 1)).dx` = dt
∴ I = `int a^t dt = a^t/loga + c`
= `(a^(x + tan^-1 x))/loga + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Evaluate :
`int(sqrt(cotx)+sqrt(tanx))dx`
Integrate the functions:
`(log x)^2/x`
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`(1+ log x)^2/x`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
`int (dx)/(sin^2 x cos^2 x)` equals:
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
Write a value of
Write a value of\[\int \cos^4 x \text{ sin x dx }\]
Write a value of\[\int\frac{\left( \tan^{- 1} x \right)^3}{1 + x^2} dx\]
Write a value of
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
The value of \[\int\frac{\cos \sqrt{x}}{\sqrt{x}} dx\] is
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals : `int(x - 2)/sqrt(x + 5).dx`
Evaluate the following integrals:
`int (sin4x)/(cos2x).dx`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
State whether the following statement is True or False.
The proper substitution for `int x(x^x)^x (2log x + 1) "d"x` is `(x^x)^x` = t
Evaluate `int "x - 1"/sqrt("x + 4")` dx
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int "x" * "e"^"2x"` dx
`int (sin4x)/(cos 2x) "d"x`
`int ("e"^(3x))/("e"^(3x) + 1) "d"x`
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int cot^2x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
`int (7x + 9)^13 "d"x` ______ + c
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 sin^-1 x`dx = ?
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
Find `int dx/sqrt(sin^3x cos(x - α))`.
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate the following.
`int(1)/(x^2 + 4x - 5)dx`
Evaluate `int 1/(x(x-1))dx`
If f'(x) = 4x3- 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4)) dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
