Advertisements
Advertisements
प्रश्न
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Advertisements
उत्तर
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
संबंधित प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`sqrt(sin 2x) cos 2x`
Write a value of\[\int\frac{\sin x + \cos x}{\sqrt{1 + \sin 2x}} dx\]
Write a value of
Write a value of\[\int\frac{\sin x - \cos x}{\sqrt{1 + \sin 2x}} \text{ dx}\]
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
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 (cos2x)/sin^2x dx`
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
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 : `((x - 1)^2)/(x^2 + 1)^2`
Integrate the following functions w.r.t. x : `(20 + 12e^x)/(3e^x + 4)`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int (1)/(1 + x - x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
`int (x^2 + x - 6)/((x - 2)(x - 1))dx = x` + ______ + c
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int cot^2x "d"x`
`int (sin (5x)/2)/(sin x/2)dx` is equal to ______. (where C is a constant of integration).
`int(log(logx) + 1/(logx)^2)dx` = ______.
If `int [log(log x) + 1/(logx)^2]dx` = x [f(x) – g(x)] + C, then ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Write `int cotx dx`.
if `f(x) = 4x^3 - 3x^2 + 2x +k, f (0) = - 1 and f (1) = 4, "find " f(x)`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate the following
`int x^3 e^(x^2) ` dx
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate `int (5x^2 - 6x + 3)/(2x - 3) dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
