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`
Integrate the function in x sin 3x.
Integrate the function in x tan-1 x.
Evaluate the following : `int sin θ.log (cos θ).dθ`
Evaluate the following : `int x.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:
`e^-x cos2x`
Integrate the following functions w.r.t. x:
sin (log x)
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
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)]`
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
`int ("x" + 1/"x")^3 "dx"` = ______
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int "dx"/("x"[(log "x")^2 + 4 log "x" - 1])`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int 1/x "d"x` = ______ + c
Evaluate `int 1/(x log x) "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`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
`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.
Find: `int e^x.sin2xdx`
`int 1/sqrt(x^2 - a^2)dx` = ______.
The integral `int x cos^-1 ((1 - x^2)/(1 + x^2))dx (x > 0)` is equal to ______.
`int(1-x)^-2 dx` = ______
`int logx dx = x(1+logx)+c`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
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^3e^(x^2)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate:
`int x^2 cos x dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
The value of `inta^x.e^x dx` equals
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?
Under the LIATE rule, which type of function has priority \(L\)?
For the special integral \[\int e^x\left[f(x)+f'(x)\right]dx,\] what is the antiderivative?
Repeated parts may be needed for which pair of integrals?
