Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Advertisements
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(x^2 - a^2)dx = x/2sqrt(x^2 - a^2) - a^2/2log|x + sqrt(x^2 - a^2)| + c`
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in x sin x.
Integrate the function in `x^2e^x`.
Integrate the function in x log 2x.
Integrate the function in x (log x)2.
Integrate the function in `(xe^x)/(1+x)^2`.
Find :
`∫(log x)^2 dx`
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following: `int logx/x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
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(2x^2 + 3x + 4)`
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^(5x).[(5x.logx + 1)/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 : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
∫ x log x dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int 1/(4x + 5x^(-11)) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int cot "x".log [log (sin "x")] "dx"` = ____________.
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
`int 1/sqrt(x^2 - 9) dx` = ______.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Find: `int e^x.sin2xdx`
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 ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate:
`int (logx)^2 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`
Evaluate the following:
`intx^3e^(x^2)dx`
Evaluate the following.
`intx^3e^(x^2) dx`
Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?
Which pair is listed as Exponential in the LIATE rule?
Using \[t=1-x^2,\] what is \[\int \frac{x\,dx}{\sqrt{1-x^2}}?\]
Which substitution can also be used before integrating by parts for \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx?\]
