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(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^2e^x`.
Integrate the function in x log 2x.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int x.cos^3x.dx`
Integrate the following functions w.r.t.x:
`e^-x cos2x`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `sec^2x.sqrt(tan^2x + tan x - 7)`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Integrate the following with respect to the respective variable : `(3 - 2sinx)/(cos^2x)`
Integrate the following with respect to the respective variable : cos 3x cos 2x cos x
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int "dx"/sqrt(4"x"^2 - 5)`
`int (cos2x)/(sin^2x cos^2x) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int logx/(1 + logx)^2 "d"x`
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
`int tan^-1 sqrt(x) "d"x` is equal to ______.
Solve: `int sqrt(4x^2 + 5)dx`
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
If `π/2` < x < π, then `intxsqrt((1 + cos2x)/2)dx` = ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
If `int (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
`intsqrt(1+x) dx` = ______
Evaluate `int(3x-2)/((x+1)^2(x+3)) dx`
`inte^(xloga).e^x dx` is ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))dx`
Evaluate:
`inte^x sinx dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3e^(x^2) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
