Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x:
sin (log x)
Advertisements
उत्तर
Le I = `int sin (logx)x dx`
Put log x = t
∴ x = et
∴ dx = et dt
∴ I = `int sin t xx e^t dt`
= `int e^t sin t dt`
= `e^t int sin t dt - int [d/dt (e^t) int sin t dt] dt`
= `e^t (- cos t) - int e^t (- cos t) dt`
= `-e^t cos t + int e^t cos t dt`
= `- e^t cos t + e^t int cos t dt - int [d/dt (e^t) int cos t dt] dt`
= `- e^t cos t + e^t sin t - int e^t sin t dt`
∴ I = – et cos t + et sin t – I
∴ 2I = et (sin t – cos t)
∴ `I = e^t/(2) (sin t - cos t) + c`
= `x/(2) [sin (logx) - cos (logx)] + c`.
APPEARS IN
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`
`int1/xlogxdx=...............`
(A)log(log x)+ c
(B) 1/2 (logx )2+c
(C) 2log x + c
(D) log x + c
Integrate the function in x sin 3x.
Integrate the function in x sin−1 x.
Integrate the function in ex (sinx + cosx).
Integrate the function in e2x sin x.
Evaluate the following: `int x.sin^-1 x.dx`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `sqrt(2x^2 + 3x + 4)`
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following with respect to the respective variable : `(sin^6θ + cos^6θ)/(sin^2θ*cos^2θ)`
Solve the following differential equation.
(x2 − yx2 ) dy + (y2 + xy2) dx = 0
Evaluate the following.
`int e^x (1/x - 1/x^2)`dx
Evaluate the following.
`int "e"^"x" "x - 1"/("x + 1")^3` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Choose the correct alternative from the following.
`int (("x"^3 + 3"x"^2 + 3"x" + 1))/("x + 1")^5 "dx"` =
Evaluate: `int ("ae"^("x") + "be"^(-"x"))/("ae"^("x") - "be"^(−"x"))` dx
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
`int 1/sqrt(2x^2 - 5) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int 1/x "d"x` = ______ + c
Evaluate `int 1/(x(x - 1)) "d"x`
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
The value of `int_(- pi/2)^(pi/2) (x^3 + x cos x + tan^5x + 1) dx` is
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.
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 ______.
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_0^1 x tan^-1 x dx` = ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
`int1/sqrt(x^2 - a^2) dx` = ______
`inte^(xloga).e^x dx` is ______
`int(f'(x))/sqrt(f(x)) dx = 2sqrt(f(x))+c`
Evaluate:
`intcos^-1(sqrt(x))dx`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate `int tan^-1x dx`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
If ∫(cot x – cosec2 x)ex dx = ex f(x) + c then f(x) will be ______.
Evaluate `int(1 + x + x^2/(2!))dx`.
Evaluate the following.
`intx^3/(sqrt(1 + x^4))dx`
`∫ sin^(−1)` xdx is equal to ______.
