Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Solution
Let I = `int sqrt(2x^2 + 3x + 4).dx`
= `sqrt(2) int sqrt(x^2 + 3/2 x + 2).dx`
= `sqrt(2) int sqrt((x^2 + 3/2x + 9/16) - 9/16 + 2).dx`
= `sqrt(2) int sqrt((x + 3/4)^2 + (sqrt(23)/4)^2).dx`
= `sqrt(2)[((x + 3/4))/(2) sqrt((x + 3/4)^2 + (sqrt(23)/4)^2 ) + ((23/16))/(2)log|(x + 3/4) + sqrt((x + 3/4)^2 + (sqrt(23)/4)^2)|] + c`
= `ssqrt(2)[((4x + 3)/8) sqrt(x^2 + 3/2x + 2) + (23)/(32)log|(x + 3/4) + sqrt(x^2 + 3/2x + 2)|] + c`.
APPEARS IN
RELATED QUESTIONS
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 sin-1 x.
Integrate the function in x sec2 x.
Integrate the function in tan-1 x.
Integrate the function in (x2 + 1) log x.
Integrate the function in ex (sinx + cosx).
Integrate the function in `e^x (1/x - 1/x^2)`.
`int e^x sec x (1 + tan x) dx` equals:
Evaluate the following : `int x tan^-1 x .dx`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following : `int logx/x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log 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` =
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log x)]*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : sec4x cosec2x
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int 1/x "d"x` = ______ + c
`int "e"^x x/(x + 1)^2 "d"x`
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
`int tan^-1 sqrt(x) "d"x` is equal to ______.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Find: `int e^x.sin2xdx`
`int 1/sqrt(x^2 - a^2)dx` = ______.
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`.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(1-x)^-2 dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
`int logx dx = x(1+logx)+c`
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate `int tan^-1x 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^3 e^(x^2) dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
The value of `inta^x.e^x dx` equals
Evaluate `int(1 + x + x^2/(2!))dx`.
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
