Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Advertisements
उत्तर
Let I = `int "cosec" (log x)[1 - cot (log x)].dx`
Put log x = t
∴ et
∴ dx = et .dt
∴ I = `int "cosec" t (1 - cot t).e^t dt`
= `int e^t ["cosec" t - "cosec" t cot t].dt`
= `int e^t ["cosec" t + d/dt ("cosec" t)].dt`
= `e^t "cosec" t + c ...[∵ int e^t [f(t) + f'(t)].dt = e^t f(t) + c]`
= x . cosec (log x) + c.
APPEARS IN
संबंधित प्रश्न
Integrate : sec3 x w. r. t. x.
Integrate the function in x sin 3x.
Integrate the function in `x^2e^x`.
Integrate the function in x log 2x.
Integrate the function in x sec2 x.
Find :
`∫(log x)^2 dx`
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int x.sin^2x.dx`
Evaluate the following: `int x.sin^-1 x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int cos sqrt(x).dx`
Integrate the following functions w.r.t. x : `[x/(x + 1)^2].e^x`
Integrate the following functions w.r.t. x : `log(1 + x)^((1 + x)`
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
If f(x) = `sin^-1x/sqrt(1 - x^2), "g"(x) = e^(sin^-1x)`, then `int f(x)*"g"(x)*dx` = ______.
Choose the correct options from the given alternatives :
`int (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int [sin (log x) + cos (log 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))`
Evaluate the following.
`int x^2 e^4x`dx
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" [(log "x")^2 + (2 log "x")/"x"]` dx
Choose the correct alternative from the following.
`int (("e"^"2x" + "e"^"-2x")/"e"^"x") "dx"` =
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
`int 1/(4x + 5x^(-11)) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int ("d"x)/(x - x^2)` = ______
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int [(log x - 1)/(1 + (log x)^2)]^2`dx = ?
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
`int "dx"/(sin(x - "a")sin(x - "b"))` is equal to ______.
If u and v ore differentiable functions of x. then prove that:
`int uv dx = u intv dx - int [(du)/(d) intv dx]dx`
Hence evaluate `intlog x dx`
`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.
`int x/((x + 2)(x + 3)) dx` = ______ + `int 3/(x + 3) dx`
Solve: `int sqrt(4x^2 + 5)dx`
If `int(2e^(5x) + e^(4x) - 4e^(3x) + 4e^(2x) + 2e^x)/((e^(2x) + 4)(e^(2x) - 1)^2)dx = tan^-1(e^x/a) - 1/(b(e^(2x) - 1)) + C`, where C is constant of integration, then value of a + b is equal to ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate the following.
`int (x^3)/(sqrt(1 + x^4))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.
`int x^3 e^(x^2) dx`
Evaluate the following.
`intx^2e^(4x)dx`
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
