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(x-3)sqrt(x^2+3x-18) dx`
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Integrate the functions:
`x/(9 - 4x^2)`
Integrate the functions:
`(1+ log x)^2/x`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of
Find : ` int (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
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 : `e^x.log (sin e^x)/tan(e^x)`
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Integrate the following functions w.r.t.x:
cos8xcotx
Integrate the following functions w.r.t. x : sin5x.cos8x
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Evaluate the following.
`int "x"^5/("x"^2 + 1)`dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Choose the correct alternative from the following.
`int "x"^2 (3)^("x"^3) "dx"` =
Evaluate: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
Evaluate: `int 1/(2"x" + 3"x" log"x")` dx
`int (2(cos^2 x - sin^2 x))/(cos^2 x + sin^2 x)` dx = ______________
`int cos^7 x "d"x`
`int (7x + 9)^13 "d"x` ______ + c
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
If f'(x) = `x + 1/x`, then f(x) is ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
`int (x + 1)/(x(1 + xe^x)) dx` is equal to
