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
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Integrate the functions:
`xsqrt(x + 2)`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
Write a value of
Write a value of
Write a value of
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals:
`int(2)/(sqrt(x) - sqrt(x + 3)).dx`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x:
`(sinx cos^3x)/(1 + cos^2x)`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following : `int (1)/sqrt(x^2 + 8x - 20).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?
Fill in the Blank.
`int 1/"x"^3 [log "x"^"x"]^2 "dx" = "P" (log "x")^3` + c, then P = _______
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: If f '(x) = `sqrt"x"` and f(1) = 2, then find the value of f(x).
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int sqrt(x) sec(x)^(3/2) tan(x)^(3/2)"d"x`
`int (cos2x)/(sin^2x) "d"x`
To find the value of `int ((1 + logx))/x` dx the proper substitution is ______
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
`int ("d"x)/(x(x^4 + 1))` = ______.
The integral `int ((1 - 1/sqrt(3))(cosx - sinx))/((1 + 2/sqrt(3) sin2x))dx` is equal to ______.
`int (logx)^2/x dx` = ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate the following.
`int x^3/(sqrt(1+x^4))dx`
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`int1/(x^2 + 4x-5)dx`
After putting \[t=\cos x\], which integral is obtained from \[\int\sin^2x\cos^2x(\sin x)\,dx\]?
What is \[\int\frac{\sin x}{\sin(x+a)}\,dx\]?
