Advertisements
Advertisements
प्रश्न
Evaluate the following : `int sqrt((2 + x)/(2 - x)).dx`
Advertisements
उत्तर
Let I = `int sqrt((2 + x)/(2 - x)).dx`
= `int sqrt((2 + x)/(2 - x) xx (2 + x)/(2 + x)).dx`
= `int (2 + x)/sqrt(4 - x^2).dx`
= `int (2)/sqrt(4 - x^2).dx + int x/sqrt(4 - x^2).dx`
= `2 int (1)/sqrt(2^2 - x^2).dx + (1)/(2) int (2x)/sqrt(4 - x^2).dx`
= I1 + I2 ...(Let)
I1 = `2 int (1)/sqrt(2^2 - x^2).dx`
= `2 sin^-1 (x/2) + c_1`
In I2, put 4 – x2 = t
∴ – 2x dx = dt
∴ 2x dx = – dt
I2 = `-(1)/(2) int t^(-1/2) dt`
= `-(1)/(2).t^(1/2)/((1/2)) + c_2`
= `- sqrt(4 - x^2) + c_2`
I = `2 sin^-1 (x/2) - sqrt(4 - x^2) + c`.
APPEARS IN
संबंधित प्रश्न
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
sin (ax + b) cos (ax + b)
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`sin x/(1+ cos x)`
Integrate the functions:
`1/(1 + cot x)`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Evaluate the following integrals : `int (sin2x)/(cosx)dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t. x:
`x^5sqrt(a^2 + x^2)`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Evaluate the following : `int (1)/sqrt(8 - 3x + 2x^2).dx`
Evaluate the following : `int (1)/(4 + 3cos^2x).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 `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Evaluate: `int 1/(sqrt("x") + "x")` dx
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
`int logx/x "d"x`
`int 1/(xsin^2(logx)) "d"x`
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
General solution of `(x + y)^2 ("d"y)/("d"x) = "a"^2, "a" ≠ 0` is ______. (c is arbitrary constant)
`int (f^'(x))/(f(x))dx` = ______ + c.
Evaluate the following.
`int 1/(x^2 + 4x - 5)dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate `int (1)/(x(x - 1))dx`
Evaluate:
`int sqrt((a - x)/x) 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 e^(x^2) dx`
Evaluate `int1/(x(x-1))dx`
Evaluate:
`intsqrt(sec x/2 - 1)dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
Which substitution is appropriate for \[\sqrt{\frac{x}{a-x}}\], \[\sqrt{\frac{a-x}{x}}\], \[\sqrt{x(a-x)}\], or \[\frac{1}{\sqrt{x(a-x)}}\]?
After choosing \[u=g(x)\], what is the next step?
