Advertisements
Advertisements
प्रश्न
Prove that:
`int sqrt(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Advertisements
उत्तर
Let I = `int sqrt(x^2 - a^2)dx`
I = `int sqrt(x^2 - a^2)*1dx`
I = `sqrt(x^2 - a^2)*int1dx - int[d/dx(sqrt(x^2 - a^2))*int1dx]dx`
I = `sqrt(x^2 - a^2)*x - int[1/(2sqrt(x^2 - a^2))*d/dx(x^2 - a^2)*x]dx`
I = `sqrt(x^2 - a^2)*x - int1/(2sqrt(x^2 - a^2))(2x - 0)*x dx`
I = `sqrt(x^2 - a^2)*x - intx/sqrt(x^2 - a^2)*x dx`
I = `xsqrt(x^2 - a^2) - int(x^2 - a^2 + a^2)/(sqrt(x^2 - a^2))dx`
I = `xsqrt(x^2 - a^2) - intsqrt(x^2 - a^2) dx - a^2intdx/(sqrt(x^2 - a^2)`
I = `xsqrt(x^2 - a^2) - I - a^2log|x + sqrt(x^2 - a^2)| + c_1`
∴ 2I = `xsqrt(x^2 - a^2) - a^2log|x + sqrt(x^2 - a^2)| + c_1`
∴ I = `x/2sqrt(x^2-a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c_1/2`
∴ `intsqrt(x^2 - a^2) dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c, "where" c = c_1/2`
APPEARS IN
संबंधित प्रश्न
Integrate : sec3 x w. r. t. x.
Integrate the function in x log x.
Integrate the function in x tan-1 x.
Integrate the function in x (log x)2.
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `e^x .(1/x - 1/x^2)`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
`int (cos2x)/(sin^2x cos^2x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int ("d"x)/(x - x^2)` = ______
`int(x + 1/x)^3 dx` = ______.
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
Evaluate `int 1/(x(x - 1)) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
`int 1/sqrt(x^2 - 9) dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
`int1/sqrt(x^2 - a^2) dx` = ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))dx`
If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv dx - int(d/dx u)(intv dx)dx`. Hence evaluate: `intx cos x dx`
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
