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 : `int (sinx)/sqrt(36-cos^2x)dx`
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Evaluate :
`∫(x+2)/sqrt(x^2+5x+6)dx`
Integrate the functions:
`x/(sqrt(x+ 4))`, x > 0
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Write a value of
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
The value of \[\int\frac{1}{x + x \log x} dx\] is
Prove that: `int "dx"/(sqrt("x"^2 +"a"^2)) = log |"x" +sqrt("x"^2 +"a"^2) | + "c"`
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals:
`int x/(x + 2).dx`
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` 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 ______.
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
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 1/(sqrt("x") + "x")` dx
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int x^x (1 + logx) "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int (cos x)/(1 - sin x) "dx" =` ______.
The general solution of the differential equation `(1 + y/x) + ("d"y)/(d"x)` = 0 is ______.
`int(log(logx) + 1/(logx)^2)dx` = ______.
`int 1/(sinx.cos^2x)dx` = ______.
Evaluate `int(1 + x + x^2/(2!))dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int1/(x(x-1))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).
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int(1 + x + x^2 / (2!))dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
