Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
उत्तर
Let I = `int (1)/(3 + 2sin x - cosx)dx`
Put `tan(x/2)` = t
∴ x = 2 tan–1 t
∴ dx = `(2)/(1 + t^2)dt` and
sinx = `(2t)/(1 + t^2)' cosx = (1 - t^2)/(1 + t^2)`
∴ I = `int (1)/(3 + 2((2t)/(1 + t^2)) - ((1 - t^2)/(1 + t^2))).(2dt)/(1 + t^2)`
= `int (1 + t^2)/(3(1 + t^2) + 4t - (1 - t^2)).(2dt)/(1 + t^2)`
= `2 int dt/(4t^2 + 4t + 2)`
= `2 int dt/(4t^2 + 4t + 1 + 1)`
= `2 int dt/((2t + 1)^2 + 1^2)`
= `(2)/(2)tan^-1((2t + 1)/1) + c`
= `tan^-1[2tan^-1(x/2) + 1] + c`.
APPEARS IN
संबंधित प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
cot x log sin x
Integrate the functions:
`1/(1 - tan x)`
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
Write a value of
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Evaluate : `int ("e"^"x" (1 + "x"))/("cos"^2("x""e"^"x"))"dx"`
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Integrate the following functions w.r.t. x : `(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*dx` =
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
`int logx/(log ex)^2*dx` = ______.
Choose the correct options from the given alternatives :
`int (e^(2x) + e^-2x)/e^x*dx` =
Integrate the following with respect to the respective variable:
`x^7/(x + 1)`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
Evaluate:
`int (5x^2 - 6x + 3)/(2x − 3)` dx
Evaluate `int "x - 1"/sqrt("x + 4")` dx
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
Evaluate: `int log ("x"^2 + "x")` dx
Evaluate: `int sqrt("x"^2 + 2"x" + 5)` dx
`int 1/(cos x - sin x)` dx = _______________
`int logx/x "d"x`
`int cot^2x "d"x`
`int (7x + 9)^13 "d"x` ______ + c
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int_1^3 ("d"x)/(x(1 + logx)^2)` = ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
`int (logx)^2/x dx` = ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
`int dx/((x+2)(x^2 + 1))` ...(given)
`1/(x^2 +1) dx = tan ^-1 + c`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate:
`int sin^3x cos^3x dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
Evaluate `int1/(x(x - 1))dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
