Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Advertisements
उत्तर
Let I = `int (1)/(2sin 2x - 3)dx`
Put tan x = t
∴ x = tan–1 t
∴ dx = `dt/(1 + t^2) and sin 2x = (2t)/(1 + t^2)`
∴ I = `int(1)/(2((2t)/(1 + t^2)) - 3).dt/(1 + t^2)`
= `int (1 + t^2)/(4t - 3 - 3t^2).dt/(1 + t^2)`
= `int (1)/(-3t^2 + 4t - 3)dt`
= `(1)/(3) int (1)/(t^2 - 4/3t + 1)dt`
= `-(1)/(3) int (1)/((t^2 - 4/3t + 4/9) - (4)/(9) + 1)dt`
= `-(1)/(3) int (1)/((t - 2/3)^2 + (sqrt(5)/3)^2)dt`
= `-(1)/(3) xx (1)/((sqrt(5)/3))tan^-1 ((t - 2/3)/(sqrt(5)/3)) + c`
= `-(1)/sqrt(5)tan^-1 ((3t - 2)/sqrt(5)) + c`
= `-(1)/sqrt(5)tan^-1((3tan x - 2)/(sqrt(5))) + 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 : `int(x-3)sqrt(x^2+3x-18) dx`
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Find `intsqrtx/sqrt(a^3-x^3)dx`
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
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 \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(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 : e3logx(x4 + 1)–1
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x : `(sin6x)/(sin 10x sin 4x)`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*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).
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int (3"e"^"x" + 4)/(2"e"^"x" - 8)`dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
If f '(x) = `1/"x" + "x"` and f(1) = `5/2`, then f(x) = log x + `"x"^2/2` + ______
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
Evaluate: `int "e"^sqrt"x"` dx
`int 1/(cos x - sin x)` dx = _______________
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int cos sqrtx` dx = _____________
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int ("d"x)/(x(x^4 + 1))` = ______.
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
Write `int cotx dx`.
`int (logx)^2/x 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)
Evaluate the following
`int1/(x^2 +4x-5)dx`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Solve the following Evaluate.
`int(5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate the following.
`int1/(x^2+4x-5) dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
In \[\int\sin^3x\cos^2x\,dx\], which rewriting prepares the integrand for the substitution \[t=\cos x\]?
