Advertisements
Advertisements
Question
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Advertisements
Solution
Let I = `int sqrt((10 + x)/(10 - x)).dx`
= `int sqrt((10 + x)/(10 - x) xx (10 + x)/(10 + x)).dx`
= `int (10 + x)/sqrt(100 - x^2).dx`
= `int (10)/sqrt(100 - x^2).dx + int x/sqrt(100 - x^2).dx`
= `10 int (1)/sqrt(10^2 - x^2).dx + (1)/(2) int (2x)/sqrt(100 - x^2).dx`
= I1 + I2 ...(Let)
I1 = `10 int (1)/sqrt(10^2 - x^2).dx`
= `10 sin^-1 (x/10) + c_1`
In I2, put 100 – 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(100 - x^2) + c_2`
I = `10 sin^-1 (x/10) - sqrt(100 - x^2) + c`.
APPEARS IN
RELATED QUESTIONS
Evaluate : `int (sinx)/sqrt(36-cos^2x)dx`
Evaluate : `int(x-3)sqrt(x^2+3x-18) dx`
Integrate the functions:
`(2x)/(1 + x^2)`
Integrate the functions:
`xsqrt(1+ 2x^2)`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Evaluate : `∫1/(3+2sinx+cosx)dx`
Solve:
dy/dx = cos(x + y)
Write a value of\[\int\text{ tan x }\sec^3 x\ dx\]
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)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 : `(x^n - 1)/sqrt(1 + 4x^n)`
Integrate the following functions w.r.t. x : tan5x
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Choose the correct options from the given alternatives :
`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int 1/(x(x^6 + 1))` dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` 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: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int log ("x"^2 + "x")` dx
Evaluate: `int "e"^sqrt"x"` dx
`int x^3"e"^(x^2) "d"x`
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
`int(7x - 2)^2dx = (7x -2)^3/21 + c`
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
Write `int cotx dx`.
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
Evaluate `int_-a^a f(x) dx`, where f(x) = `9^x/(1 + 9^x)`.
Evaluate `int 1/("x"("x" - 1)) "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 sqrt(1 + x^2) 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 `int (1 + "x" + "x"^2/(2!))`dx
Evaluate:
`intsqrt(sec x/2 - 1)dx`
What is \[\int\cot t\,dt\] in the evaluation of \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
