Advertisements
Advertisements
प्रश्न
Integrate the following w.r.t.x : sec4x cosec2x
Advertisements
उत्तर
Let I = `int sec^4x "cosec"^2x*dx`
= `int sec^4x "cosec"^2x* sec^2x*dx`
Put tan x = t
∴ sec2x·dx = d
Also, sec2x cosec2x = (1 + tan2x)(1 + cot2x)
= `(1 + t^2)(1 + 1/t^2)`
= `(1 + t^2)((t^2 + 1)/t^2)`
= `(t^4 + 2t^2 + 1)/t^2`
= `t^2 + 2 + (1)/t^2`
∴ I = `int (t^2 + 2 + 1/t^2)*dt`
= `int t^2*dt + 2 int *dt + int 1/t^2*dt`
= `t^3/(3) + 2t + (t^-1)/((-1)) + c`
= `(1)/(3)tan^3x + 2tanx - (1)/tanx + c`
= `(1)/(3cot^3x) + (2)/(cotx) - cot x + c`.
APPEARS IN
संबंधित प्रश्न
Prove that: `int sqrt(a^2 - x^2) * dx = x/2 * sqrt(a^2 - x^2) + a^2/2 * sin^-1(x/a) + c`
Integrate the function in x2 log x.
Integrate the function in x tan-1 x.
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Find :
`∫(log x)^2 dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x.sin^2x.dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following w.r.t.x : log (log x)+(log x)–2
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (sinx)/(1 + sin x) "d"x`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int 1/sqrt(2x^2 - 5) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
`int ("d"x)/(x - x^2)` = ______
`int(x + 1/x)^3 dx` = ______.
`int"e"^(4x - 3) "d"x` = ______ + c
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Find: `int (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int(logx)^2dx` equals ______.
`int e^x [(2 + sin 2x)/(1 + cos 2x)]dx` = ______.
`int_0^1 x tan^-1 x dx` = ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
The integrating factor of `ylogy.dx/dy+x-logy=0` is ______.
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
Evaluate `int tan^-1x dx`
Evaluate:
`int (sin(x - a))/(sin(x + a))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/sqrt(1+x^4)`dx
Evaluate the following.
`intx^3 e^(x^2)dx`
Which function is an example under priority \(I\) in the LIATE rule?
Which pair is listed as Trigonometric in the LIATE rule?
Which pair is listed as Exponential in the LIATE rule?
Evaluate \[\int \frac{x\sin^{-1}x}{\sqrt{1-x^2}}\,dx.\]
Let \[I=\int e^x\sin x\,dx.\] After applying integration by parts once, which equation is obtained?
