Advertisements
Advertisements
प्रश्न
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Advertisements
उत्तर
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
संबंधित प्रश्न
Evaluate the following : `int x^2.log x.dx`
Evaluate the following:
`int x^2 sin 3x dx`
Evaluate the following : `int log(logx)/x.dx`
Evaluate the following : `int (t.sin^-1 t)/sqrt(1 - t^2).dt`
Evaluate the following : `int sin θ.log (cos θ).dθ`
Integrate the following functions w.r.t. x : `e^x/x [x (logx)^2 + 2 (logx)]`
Integrate the following w.r.t.x : sec4x cosec2x
Evaluate the following.
`int x^2 *e^(3x)`dx
Evaluate the following.
`int (log "x")/(1 + log "x")^2` dx
`int (sinx)/(1 + sin x) "d"x`
`int 1/(4x + 5x^(-11)) "d"x`
`int (cos2x)/(sin^2x cos^2x) "d"x`
`int ("d"x)/(x - x^2)` = ______
`int 1/sqrt(x^2 - 8x - 20) "d"x`
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 (f(x))/(log(sin x))dx` = log[log sin x] + c, then f(x) is equal to ______.
Find `int (sin^-1x)/(1 - x^2)^(3//2) dx`.
`int(1-x)^-2 dx` = ______
`intsqrt(1+x) dx` = ______
Solution of the equation `xdy/dx=y log y` is ______
Evaluate the following.
`int x^3 e^(x^2) dx`
`int1/(x+sqrt(x)) dx` = ______
`int logx dx = x(1+logx)+c`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x 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 ______.
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^3e^(x^2) dx`
Evaluate `int (1 + x + x^2/(2!))dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate `int(1 + x + x^2/(2!))dx`.
`∫ sin^(−1)` xdx is equal to ______.
Which expression is the integration-by-parts form for a product \(f(x)g(x)\)?
When applying integration by parts to \(\log x\) and inverse trig, what should they be multiplied by?
