Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Evaluate:
`inte^(sin-1) x ((x+sqrt(1-x^2))/(sqrt(1-x^2)))dx`
Advertisements
उत्तर
Let I = `int e^(sin^-1x)[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]*dx`
= `int e^(sin^-1x) [x + sqrt(1 - x^2)]*(1)/sqrt(1 - x^2)*dx`
Put sin–1 x = t
∴ `(1)/sqrt(1 - x^2) * dx` = dt
and x = sin t
∴ I = `int e^t [sin t + sqrt(1 - sin^2 t)]*dt`
= `int e^t [sin t + sqrt(cos^2t)]*dt`
= `int e^t(sin t + cos t)*dt`
Let f(t) = sin t
∴ f'(t) = cos t
∴ I = `int e^t[f(t) + f'(t)]*dt`
= et . f(t) + c
= et . sin t + c
= `e^(sin^(–1)x) * x + c`
= `x * e^(sin^(-1)x) + c`
संबंधित प्रश्न
Integrate the function in x sin−1 x.
Integrate the function in tan-1 x.
Integrate the function in (x2 + 1) log x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
`intx^2 e^(x^3) dx` equals:
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `xsqrt(5 - 4x - x^2)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Choose the correct options from the given alternatives :
`int (1)/(x + x^5)*dx` = f(x) + c, then `int x^4/(x + x^5)*dx` =
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*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`
Integrate the following w.r.t.x : `(1)/(xsin^2(logx)`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int 1/(4x + 5x^(-11)) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int(x + 1/x)^3 dx` = ______.
`int 1/x "d"x` = ______ + c
State whether the following statement is True or False:
If `int((x - 1)"d"x)/((x + 1)(x - 2))` = A log|x + 1| + B log|x – 2|, then A + B = 1
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
∫ log x · (log x + 2) dx = ?
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int(logx)^2dx` equals ______.
If `int(x + (cos^-1 3x)^2)/sqrt(1 - 9x^2)dx = 1/α(sqrt(1 - 9x^2) + (cos^-1 3x)^β) + C`, where C is constant of integration , then (α + 3β) is equal to ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Solution of the equation `xdy/dx=y log y` is ______
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`inte^x sinx dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Which pair is listed as Trigonometric in the LIATE rule?
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
