Advertisements
Advertisements
Questions
Integrate the following functions w.r.t. x : `e^(sin^-1x)*[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]`
Evaluate:
`inte^(sin-1) x ((x+sqrt(1-x^2))/(sqrt(1-x^2)))dx`
Solution
Let I = `int e^(sin^-1x)[(x + sqrt(1 - x^2))/sqrt(1 - x^2)]*dx`
= `int e^(sin^-1x) [x + sqrt(1 - x^2)]*(1)/sqrt(1 - x^2)*dx`
Put sin–1 x = t
∴ `(1)/sqrt(1 - x^2) * dx` = dt
and x = sin t
∴ I = `int e^t [sin t + sqrt(1 - sin^2 t)]*dt`
= `int e^t [sin t + sqrt(cos^2t)]*dt`
= `int e^t(sin t + cos t)*dt`
Let f(t) = sin t
∴ f'(t) = cos t
∴ I = `int e^t[f(t) + f'(t)]*dt`
= et . f(t) + c
= et . sin t + c
= `e^(sin^(–1)x) * x + c`
= `x * e^(sin^(-1)x) + c`
RELATED QUESTIONS
Integrate : sec3 x w. r. t. x.
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 cos-1 x.
Integrate the function in x sec2 x.
Integrate the function in (x2 + 1) log x.
Integrate the function in ex (sinx + cosx).
Integrate the function in `(xe^x)/(1+x)^2`.
Integrate the function in `e^x (1/x - 1/x^2)`.
Integrate the function in `((x- 3)e^x)/(x - 1)^3`.
Integrate the function in e2x sin x.
Prove that:
`int sqrt(x^2 + a^2)dx = x/2 sqrt(x^2 + a^2) + a^2/2 log |x + sqrt(x^2 + a^2)| + c`
Evaluate the following : `int x^3.tan^-1x.dx`
Evaluate the following : `int x^3.logx.dx`
Evaluate the following : `int x^2*cos^-1 x*dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int x.cos^3x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `e^(2x).sin3x`
Integrate the following functions w.r.t. x : `x^2 .sqrt(a^2 - x^6)`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x : [2 + cot x – cosec2x]ex
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Choose the correct options from the given alternatives :
`int (sin^m x)/(cos^(m+2)x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Choose the correct options from the given alternatives :
`int cos -(3)/(7)x*sin -(11)/(7)x*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 : `t^3/(t + 1)^2`
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
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : `(1)/(x^3 sqrt(x^2 - 1)`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int "x"^2 "e"^"4x"`dx
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
Evaluate the following.
`int "e"^"x" (1/"x" - 1/"x"^2)`dx
Evaluate the following.
`int "e"^"x" "x"/("x + 1")^2` dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: Find the primitive of `1/(1 + "e"^"x")`
Evaluate: `int e^x/sqrt(e^(2x) + 4e^x + 13)` dx
Evaluate: `int "dx"/(5 - 16"x"^2)`
Evaluate: `int "e"^"x"/(4"e"^"2x" -1)` dx
`int 1/(4x + 5x^(-11)) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
Choose the correct alternative:
`int ("d"x)/((x - 8)(x + 7))` =
`int 1/x "d"x` = ______ + c
`int 1/(x^2 - "a"^2) "d"x` = ______ + c
`int (x^2 + x - 6)/((x - 2)(x - 1)) "d"x` = x + ______ + c
Evaluate `int (2x + 1)/((x + 1)(x - 2)) "d"x`
`int 1/sqrt(x^2 - 8x - 20) "d"x`
`int_0^"a" sqrt("x"/("a" - "x")) "dx"` = ____________.
`int "e"^x [x (log x)^2 + 2 log x] "dx"` = ______.
`int "e"^x int [(2 - sin 2x)/(1 - cos 2x)]`dx = ______.
Find `int_0^1 x(tan^-1x) "d"x`
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int_0^1 x log(1 + 2x) "d"x`
Evaluate the following:
`int_0^pi x log sin x "d"x`
`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.
Evaluate: `int_0^(pi/4) (dx)/(1 + tanx)`
`int 1/sqrt(x^2 - a^2)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 ______.
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 ______.
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 ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
`int1/sqrt(x^2 - a^2) dx` = ______
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
`inte^(xloga).e^x dx` is ______
`int(xe^x)/((1+x)^2) dx` = ______
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`
The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.
Evaluate the following.
`intx^3 e^(x^2) 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:
`int1/(x^2 + 25)dx`
Evaluate the following.
`intx^3/sqrt(1+x^4) dx`
If f'(x) = 4x3 - 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate the following.
`int x^3 e^(x^2) dx`
Evaluate `int(1 + x + x^2/(2!))dx`.
