Advertisements
Advertisements
प्रश्न
`int sqrt(tanx) + sqrt(cotx) "d"x`
Advertisements
उत्तर
Let I = `int (sqrt(tanx) + sqrt(cotx)) "d"x`
= `int (sqrt(tanx) + 1/sqrt(tanx)) "d"x`
= `int (tanx + 1)/sqrt(tanx) "d"x`
Put `sqrt(tanx)` = t
∴ tanx = t2
∴x = tan−1(t2)
∴ dx = `1/(1 + ("t"^2)^2) * 2"t" "dt"`
∴ dx = `(2"t")/(1 + "t"^4) "dt"`
∴ I = `int ("t"^2 + 1)/"t"* (2"t")/(1 + "t"^4) "dt"`
= `2 int ("t"^2 + 1)/("t"^4 + 1) "dt"`
= `2 int (1 + 1/"t"^2)/("t"^2 + 1/"t"^2) "dt"`
= `2 int (1 + 1/"t"^2)/(("t" - 1/"t")^2 + 2)`
Put `"t" - 1/"t"` = u
∴ `(1 + 1/"t"^2) "dt"` = du
∴ I = `2 int "du"/("u"^2 + 2)`
= `2 int "du"/("u"^2 + (sqrt(2))^2`
= `2* 1/sqrt(2)tan^-1 ("u"/sqrt(2)) + "c"`
= `sqrt(2)tan^-1 (("t" - 1/"t")/sqrt(2)) + "c"`
= `sqrt(2)tan^-1 (("t"^2 - 1)/sqrt(2)) + "c"`
= `sqrt(2)tan^-1 ((tanx - 1)/sqrt(2tanx)) + "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`
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
If u and v are two functions of x then prove that
`intuvdx=uintvdx-int[du/dxintvdx]dx`
Hence evaluate, `int xe^xdx`
Evaluate `int_0^(pi)e^2x.sin(pi/4+x)dx`
Integrate the function in `x^2e^x`.
Integrate the function in tan-1 x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Integrate the function in e2x sin x.
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following:
`int x tan^-1 x . dx`
Evaluate the following : `int x^2tan^-1x.dx`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - 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)`.
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
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 (log (3x))/(xlog (9x))*dx` =
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Evaluate the following.
`int "x"^2 *"e"^"3x"`dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/("9x"^2 - 25)`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int "e"^x x/(x + 1)^2 "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int(logx)^2dx` equals ______.
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((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Evaluate the following.
`int x^3 e^(x^2) dx`
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
`int1/(x+sqrt(x)) dx` = ______
`int logx dx = x(1+logx)+c`
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^3e^(x^2) dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate the following.
`intx^3 e^(x^2)dx`
