Advertisements
Advertisements
प्रश्न
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
Advertisements
उत्तर
Let I = `int ["cosec"(logx)][1 - cot(logx)] "d"x`
Put logex = t
∴ x = et
∴ dx = `"e"^"t"*"dt"`
∴ I = `int "cosec" "t"(1 - cot "t") "e"^"t" "dt"`
= `int "e"^"t" ("cosec" "t" - "cosec" "t"*cot "t") "dt"`
Put f(t) = cosec t
∴ f'(t) = −cosec t.cot t
∴ I = `int"e"^"t" ["f"("t") + "f'"("t")] "dt"`
= et ⋅ f(t) + c = et cosec t + c
∴ I = `x "cosec" (logx) + "c"`
संबंधित प्रश्न
Integrate : sec3 x w. r. t. x.
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 sec2 x.
Integrate the function in (x2 + 1) log x.
Integrate the function in `(xe^x)/(1+x)^2`.
Evaluate the following : `int x^2tan^-1x.dx`
Evaluate the following:
`int sec^3x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int e^(2x).cos 3x.dx`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Choose the correct options from the given alternatives :
`int (log (3x))/(xlog (9x))*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Integrate the following w.r.t.x : `sqrt(x)sec(x^(3/2))*tan(x^(3/2))`
Integrate the following w.r.t.x : `log (1 + cosx) - xtan(x/2)`
Integrate the following w.r.t.x : log (x2 + 1)
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
`int 1/sqrt(2x^2 - 5) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int log x * [log ("e"x)]^-2` dx = ?
Evaluate the following:
`int_0^pi x log sin x "d"x`
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
`int 1/sqrt(x^2 - 9) dx` = ______.
Find: `int e^x.sin2xdx`
Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
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 ______.
Evaluate :
`int(4x - 6)/(x^2 - 3x + 5)^(3/2) dx`
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
`int logx dx = x(1+logx)+c`
Solve the following
`int_0^1 e^(x^2) x^3 dx`
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^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
