Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Advertisements
उत्तर
Let I = `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Put tan x = t
∴ x = tan–1 t
∴ dx = `dt/(1 + t^2) and sin 2x = (2t)/(1 + t^2),, cos2x = (1 - t^2)/(1 + t^2)`
∴ I = `int (1)/(3 + 2((2t)/(1 + t^2)) + 4((1 - t^2)/(1 + t^2))).dt/(1 + t^2)`
= `int (1 + t^2)/(3(1 + t^2) + 4t + 4(1 - t^2)).dt/(1 + t^2)`
= `int (1)/(7 + 4t - t^2)dt = int (1)/(7 - (t^2 - 4t + 4) + 4)dt`
= `int (1)/((sqrt(11))^2 - (t - 2)^2)dt`
= `(1)/(2sqrt(11))log|(sqrt(11) + t - 2)/(sqrt(11) - t + 2)| + c`
= `(1)/(2sqrt(11))log|(sqrt(11) + tan x - 2)/(sqrt(11) - tan x + 2)| + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Evaluate : `int_0^pi(x)/(a^2cos^2x+b^2sin^2x)dx`
Integrate the functions:
sin x ⋅ sin (cos x)
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
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
Integrate the following w.r.t. x : x3 + x2 – x + 1
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals : `int tanx/(sec x + tan x)dx`
Evaluate the following integrals : `int (3)/(sqrt(7x - 2) - sqrt(7x - 5)).dx`
If `f'(x) = x - (3)/x^3, f(1) = (11)/(2)`, find f(x)
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^(3x)/(e^(3x) + 1)`
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
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 : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/(x^2 + 8x + 12).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
Evaluate `int (5"x" + 1)^(4/9)` dx
`int (sin4x)/(cos 2x) "d"x`
`int cot^2x "d"x`
`int(log(logx))/x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "d"x`
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
`int(sin2x)/(5sin^2x+3cos^2x) dx=` ______.
`int ("d"x)/(x(x^4 + 1))` = ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate `int1/(x(x-1))dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
