Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Advertisements
उत्तर
Let I = `int sqrt(x^2 + 2x + 5).dx`
= `int sqrt(x^2 + 2x + 1 + 4)dx`
= `int sqrt((x + 1)^2 + 2^2).dx`
= `((x + 1)/2) int sqrt((x + 1)^2 + 2^2) + 2^2/(2)log|(x + 1) + sqrt((x + 1)^2 + 2^2)| + c`
= `((x + 1)/2)sqrt(x^2 + 2x + 5) + 2log|(x + 1) + sqrt(x^2 + 2x + 5)| + 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`
Integrate the function in x sec2 x.
Integrate the function in (x2 + 1) log x.
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int(sin(logx)^2)/x.log.x.dx`
Evaluate the following: `int logx/x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
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)]`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Choose the correct options from the given alternatives :
`int sin (log 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 w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
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"/("x + 1")^2` dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` 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"/(5 - 16"x"^2)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int(x + 1/x)^3 dx` = ______.
`int 1/x "d"x` = ______ + c
Evaluate `int 1/(x(x - 1)) "d"x`
Solve: `int sqrt(4x^2 + 5)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 ______.
Evaluate:
`int(1+logx)/(x(3+logx)(2+3logx)) dx`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`intcos^-1(sqrt(x))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate the following.
`intx^3 e^(x^2) dx`
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 the following.
`intx^3 e^(x^2) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate `int(1 + x + x^2/(2!))dx`.
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Which substitution can also be used before integrating by parts for \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx?\]
Evaluate \[\int x\cos x\,dx.\]
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]
