Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Advertisements
उत्तर
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
संबंधित प्रश्न
`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`
Integrate the function in x log x.
Integrate the function in tan-1 x.
Integrate the function in (x2 + 1) log x.
Integrate the function in `e^x (1/x - 1/x^2)`.
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 x.sin^-1 x.dx`
Evaluate the following : `int x.cos^3x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
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` =
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : e2x sin x cos x
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
Evaluate: `int "dx"/(5 - 16"x"^2)`
`int 1/(4x + 5x^(-11)) "d"x`
`int 1/x "d"x` = ______ + c
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int"e"^(4x - 3) "d"x` = ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
The value of `int "e"^(5x) (1/x - 1/(5x^2)) "d"x` is ______.
Find `int_0^1 x(tan^-1x) "d"x`
`int 1/sqrt(x^2 - 9) 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.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Solution of the equation `xdy/dx=y log y` is ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`inte^x sinx dx`
Under the LIATE rule, which type of function has priority \(L\)?
Which function is an example under priority \(I\) in the LIATE rule?
Which pair is listed as Exponential in the LIATE rule?
Evaluate \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx.\]
Evaluate \[\int e^x\sin x\,dx.\]
Which differentiation identity verifies the special integral \[\int e^x\left[f(x)+f'(x)\right]dx=e^xf(x)+C?\]
