Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
उत्तर
Let I = `int sqrt((x - 3)(7 - x)).dx`
=`intsqrt(-x^2 + 10x - 21).dx`
= `int sqrt(- (x^2 - 10x + 21)).dx`
= `int sqrt(4 -(x^2 - 10x + 25)).dx`
= `int sqrt(2^2 - (x - 5)^2`
= `((x - 5)/2) sqrt(2^2 - (x - 5)^2) + 2^2/(2) sin^-1 ((x - 5)/2) + c`
= `((x - 5)/2) sqrt((x - 3)(7 - x)) + 2sin^-1 ((x - 5)/2) + 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`
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin 3x.
Integrate the function in x log x.
`int e^x sec x (1 + tan x) dx` equals:
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int logx/x.dx`
Evaluate the following:
`int x.sin 2x. cos 5x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Integrate the following functions w.r.t. x : `sqrt(x^2 + 2x + 5)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)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 : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
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 (1 + cosx) - xtan(x/2)`
Evaluate the following.
`int "x"^2 "e"^"4x"`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int [1/(log "x") - 1/(log "x")^2]` 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: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`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` = ______.
Evaluate `int 1/(x(x - 1)) "d"x`
Find `int_0^1 x(tan^-1x) "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
`int 1/sqrt(x^2 - 9) dx` = ______.
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 ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int_0^1 x tan^-1 x dx` = ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate:
`inte^x sinx dx`
Evaluate:
`int (logx)^2 dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
Complete the following activity:
`int_0^2 dx/(4 + x - x^2) `
= `int_0^2 dx/(-x^2 + square + square)`
= `int_0^2 dx/(-x^2 + x + 1/4 - square + 4)`
= `int_0^2 dx/ ((x- 1/2)^2 - (square)^2)`
= `1/sqrt17 log((20 + 4sqrt17)/(20 - 4sqrt17))`
Evaluate:
`int1/(x^2 + 25)dx`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
