Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Advertisements
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
Integrate the functions:
`xsqrt(x + 2)`
`(10x^9 + 10^x log_e 10)/(x^10 + 10^x) dx` equals:
Evaluate: `int 1/(x(x-1)) dx`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`
Evaluate: `int_0^3 f(x)dx` where f(x) = `{(cos 2x, 0<= x <= pi/2),(3, pi/2 <= x <= 3) :}`
Write a value of
Write a value of
Write a value of\[\int\frac{1}{1 + e^x} \text{ dx }\]
Write a value of\[\int \log_e x\ dx\].
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Evaluate the following integrals:
`int (cos2x)/sin^2x dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
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 : e3logx(x4 + 1)–1
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Integrate the following functions w.r.t. x : `(cos3x - cos4x)/(sin3x + sin4x)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
Evaluate `int (1 + x + x^2/(2!))`dx
Evaluate `int 1/(x (x - 1))` dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int x/(4x^4 - 20x^2 - 3) dx`
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
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))`=
`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
`int sqrt(1 + sin2x) dx`
`int sec^6 x tan x "d"x` = ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
Find : `int sqrt(x/(1 - x^3))dx; x ∈ (0, 1)`.
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int(5x^2-6x+3)/(2x-3)dx`
