Advertisements
Advertisements
Question
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Advertisements
Solution
Let I = `int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Put x = sin θ
⇒ dx = cos θ dθ
I = `int (sin^-1(sin theta))/((1 - sin^2 theta)^(3/2)) * cos theta "d"theta`
= `int (theta * cos theta "d"theta)/((cos^2 theta)^(3/2))`
= `int (theta * cos theta)/(cos^3 theta) "d"theta`
= `int theta/(cos^2 theta) "d"theta`
= `int theta_"I" sec_"II"^2theta "d"theta`
=`theta * sec^2theta "d"theta - int ("D"(theta) * int sec^2theta "d"theta)"d"theta` .....`[because int "u"_"I" * "v"_"II" "d"x = "u" * int "v" "d"x - int ("D"("u") int "v" "dv")"dv" + "C"]`
= `theta * tan theta - int 1 * tan theta "d"theta`
= `theta * tan theta - log sec theta + "C"`
= `sin^-1x * x/sqrt(1 - x^2) - log|sqrt(1 - x^2)| + "C"` ......`[("When" x = sin theta),(therefore tan theta = x/sqrt(1 - x^2) "and" sec theta = sqrt(1 - x^2))]`
Hence, I = `(x sin^-1x)/sqrt(1 - x^2) - log|sqrt(1 - x^2)| + "C"`
APPEARS IN
RELATED QUESTIONS
If `int_(-pi/2)^(pi/2)sin^4x/(sin^4x+cos^4x)dx`, then the value of I is:
(A) 0
(B) π
(C) π/2
(D) π/4
Integrate the function in `x^2e^x`.
Integrate the function in x2 log x.
Integrate the function in x cos-1 x.
Integrate the function in `(xe^x)/(1+x)^2`.
Evaluate the following : `int x^2.log x.dx`
Evaluate the following : `int cos(root(3)(x)).dx`
Integrate the following functions w.r.t. x : `sqrt((x - 3)(7 - x)`
Integrate the following functions w.r.t. x : `(x + 1) sqrt(2x^2 + 3)`
Integrate the following functions w.r.t. x : `((1 + sin x)/(1 + cos x)).e^x`
Choose the correct options from the given alternatives :
`int tan(sin^-1 x)*dx` =
Choose the correct options from the given alternatives :
`int (x- sinx)/(1 - cosx)*dx` =
Integrate the following w.r.t. x: `(1 + log x)^2/x`
Integrate the following w.r.t.x : log (x2 + 1)
Integrate the following w.r.t.x : e2x sin x cos x
Evaluate the following.
∫ x log x dx
Evaluate the following.
`int [1/(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"` =
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int sqrt(tanx) + sqrt(cotx) "d"x`
Choose the correct alternative:
`intx^(2)3^(x^3) "d"x` =
`int 1/x "d"x` = ______ + c
`int "e"^x x/(x + 1)^2 "d"x`
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 (2x)/((x^2 + 1)(x^2 + 2)) dx`
`int(logx)^2dx` equals ______.
`int((4e^x - 25)/(2e^x - 5))dx = Ax + B log(2e^x - 5) + c`, then ______.
Find `int e^x ((1 - sinx)/(1 - cosx))dx`.
Find: `int e^(x^2) (x^5 + 2x^3)dx`.
`int(3x^2)/sqrt(1+x^3) dx = sqrt(1+x^3)+c`
Evaluate:
`int e^(ax)*cos(bx + c)dx`
Evaluate:
`int e^(logcosx)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`
Evaluate `int tan^-1x dx`
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 the following.
`intx^3e^(x^2) dx`
