Advertisements
Advertisements
प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Advertisements
उत्तर
`Let I =int1/(x^2sqrt(a^2+x^2)dx`
`Put x = a tantheta`
Differentiating w.r.t. theta we get
`dx = a sec^2 theta d theta`
`theta=tan^-1(x/a)`
`I=int(asec^2theta d theta)/(a^2tan^2thetasqrt(a^2+a^2tan^2theta))`
`=1/a^2intsectheta/tan^2theta d theta`
`=1/a^2intcostheta/sin^2thetad theta`
`=1/a^2intcosecthetacotthetad theta`
`I=-1/a^2cosectheta+c ....(i)`
`But tantheta=x/a`
`cottheta`=a/x`
`cosec^2theta`=1+cot^2theta`
`cosec^2theta=1+a^2/x^2`
`cosec^2theta=(x^2+a^2)/x^2`
`cosectheta=sqrt(x^2_a^2)/x.........(ii)`
`I=-1/a^2sqrt(x^2+a^2)/x+c `
APPEARS IN
संबंधित प्रश्न
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of\[\int\frac{\sec^2 x}{\left( 5 + \tan x \right)^4} dx\]
Write a value of\[\int a^x e^x \text{ dx }\]
Write a value of
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Show that : `int _0^(pi/4) "log" (1+"tan""x")"dx" = pi /8 "log"2`
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Integrate the following functions w.r.t. x : `(1 + x)/(x.sin (x + log x)`
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t.x:
cos8xcotx
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Evaluate the following.
`int (20 - 12"e"^"x")/(3"e"^"x" - 4)`dx
Evaluate the following.
`int 1/(7 + 6"x" - "x"^2)` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
`int sqrt(1 + "x"^2) "dx"` =
Evaluate: `int (2"e"^"x" - 3)/(4"e"^"x" + 1)` dx
`int x^2/sqrt(1 - x^6)` dx = ________________
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int ("e"^(2x) + "e"^(-2x))/("e"^x) "d"x`
`int (cos2x)/(sin^2x) "d"x`
`int(log(logx))/x "d"x`
State whether the following statement is True or False:
If `int x "f"(x) "d"x = ("f"(x))/2`, then f(x) = `"e"^(x^2)`
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
`int x^3"e"^(x^2) "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
Write `int cotx dx`.
Evaluate `int 1/("x"("x" - 1)) "dx"`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))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).
