Advertisements
Advertisements
प्रश्न
Integrate the function in `sin^(-1) ((2x)/(1+x^2))`.
Advertisements
उत्तर
Let `I = sin^-1 ((2x)/ (1 + x^2)) dx`
Put x = tan t
⇒ dx = sec2 t dt
∴ `I = int sin^-1 ((2 tan t)/ (1 + tan^2 t)) sec^2 t dt`
`= int sin^-1 (sin 2t) sec^2 t dt`
`= 2t sec^2 t dt = 2 int sec^2 t dt`
`= 2 {t int sec^2 t dt - int [d/dt(t) * int sec^2 t dt] dt}`
`= 2 [t tant - int 1 * tan t dt]`
= 2 t tan t + 2 log |cos t| + C
`= 2 tan^-1 x*x + 2 log |1/ sqrt (1 + x^2)| + C` `...[∵ cos t = 1/ (sect) = 1/ (sqrt (1 + tan^2 t)) = 1/ (sqrt (1 + x^2))]`
`= 2 x tan^-1 x + 2 log |(1 + x^2)^(1/2)| + C`
`= 2 x tan^-1 x + 2 (- 1/2) log |1 + x^2| + C`
`= 2 x tan^-1 x - log |1 + x^2| + C`
APPEARS IN
संबंधित प्रश्न
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`
Integrate the function in x2 log x.
Evaluate the following : `int e^(2x).cos 3x.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 : `sqrt(5x^2 + 3)`
Integrate the following functions w.r.t. x : `sqrt(4^x(4^x + 4))`
Integrate the following functions w.r.t. x: `sqrt(x^2 + 2x + 5)`.
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 (1)/(cosx - cos^2x)*dx` =
Choose the correct options from the given alternatives :
`int sin (log x)*dx` =
Integrate the following with respect to the respective variable : `t^3/(t + 1)^2`
Integrate the following w.r.t.x : cot–1 (1 – x + x2)
Evaluate the following.
`int x^2 *e^(3x)`dx
Choose the correct alternative from the following.
`int (1 - "x")^(-2) "dx"` =
Evaluate: `int "dx"/(3 - 2"x" - "x"^2)`
Evaluate: `int "dx"/(25"x" - "x"(log "x")^2)`
`int (sin(x - "a"))/(cos (x + "b")) "d"x`
`int ["cosec"(logx)][1 - cot(logx)] "d"x`
`int ("e"^xlog(sin"e"^x))/(tan"e"^x) "d"x`
`int(x + 1/x)^3 dx` = ______.
Evaluate the following:
`int (sin^-1 x)/((1 - x)^(3/2)) "d"x`
Evaluate the following:
`int ((cos 5x + cos 4x))/(1 - 2 cos 3x) "d"x`
The value of `int_0^(pi/2) log ((4 + 3 sin x)/(4 + 3 cos x)) dx` is
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`
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 the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.
`int_0^1 x tan^-1 x dx` = ______.
`int1/sqrt(x^2 - a^2) dx` = ______
`int(xe^x)/((1+x)^2) dx` = ______
Evaluate `int(1 + x + (x^2)/(2!))dx`
Evaluate:
`int((1 + sinx)/(1 + cosx))e^x dx`
`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.
Evaluate the following.
`intx^3 e^(x^2) dx`
Evaluate the following.
`intx^3/sqrt(1+x^4)`dx
Evaluate:
`inte^x "cosec" x(1 - cot x)dx`
