Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
उत्तर
Let I = `int sec^2x.sqrt(tan^2x + tan x - 7)`
Put tan x = t
∴ sec2x.dx = dt
∴ I = `int sqrt(t^2 + t - 7).dt`
= `int sqrt(t^2 + t + 1/4 - 29/4).dt`
= `int sqrt((t + 1/2)^2 - (sqrt(29)/2)^2).dt`
= `((t + 1/2)/2) sqrt((t + 1/2)^2 - 29/4) - ((29/4))/(2)log|(t + 1/2) + sqrt((t + 1/2)^2 - 29/4)| + c`
= `((2t + 1))/(4)sqrt(t^2 + t - 7) - (29)/(8)log|(t + 1/2) + sqrt(t^2 + t - 7)| + c`
= `((2tanx + 1)/4)sqrt(tan^2x + tanx - 7) - (29)/(8)log|(tanx + 1/2) + sqrt(tan^2x + tanx - 7)| + c`.
APPEARS IN
संबंधित प्रश्न
Prove that:
`int sqrt(a^2 - x^2) dx = x/2 sqrt(a^2 - x^2) + a^2/2sin^-1(x/a)+c`
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`
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in `(x cos^(-1) x)/sqrt(1-x^2)`.
Integrate the function in tan-1 x.
Integrate the function in x (log x)2.
Integrate the function in `e^x (1 + sin x)/(1+cos x)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
`intx^2 e^(x^3) dx` equals:
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `sqrt(x^2 + 2x + 5)`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
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 (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t.x : e2x sin x cos x
Integrate the following w.r.t.x : sec4x cosec2x
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "e"^"x" (1/"x" - 1/"x"^2)`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int(x + 1/x)^3 dx` = ______.
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = 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
Evaluate `int 1/(x(x - 1)) "d"x`
Evaluate `int 1/(x log x) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int "e"^x x/(x + 1)^2 "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
∫ log x · (log x + 2) dx = ?
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
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`
State whether the following statement is true or false.
If `int (4e^x - 25)/(2e^x - 5)` dx = Ax – 3 log |2ex – 5| + c, where c is the constant of integration, then A = 5.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
Solve: `int sqrt(4x^2 + 5)dx`
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find `int e^(cot^-1x) ((1 - x + x^2)/(1 + x^2))dx`.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Solution of the equation `xdy/dx=y log y` is ______
Solve the differential equation (x2 + y2) dx - 2xy dy = 0 by completing the following activity.
Solution: (x2 + y2) dx - 2xy dy = 0
∴ `dy/dx=(x^2+y^2)/(2xy)` ...(1)
Puty = vx
∴ `dy/dx=square`
∴ equation (1) becomes
`x(dv)/dx = square`
∴ `square dv = dx/x`
On integrating, we get
`int(2v)/(1-v^2) dv =intdx/x`
∴ `-log|1-v^2|=log|x|+c_1`
∴ `log|x| + log|1-v^2|=logc ...["where" - c_1 = log c]`
∴ x(1 - v2) = c
By putting the value of v, the general solution of the D.E. is `square`= cx
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
