Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(2 + cosx - sinx).dx`
Advertisements
उत्तर
Let I = `int (1)/(2 + cosx - sinx).dx`
Put `tan (x/2)` = t
∴ x 2 tan–1 t
∴ dx = `(2dt)/(1 + t^2) and sin x = (2t)/(1 + t^2), cosx = (1 - t^2)/(1 + t^2)`
∴ I = `int (1)/(2 + ((1 - t^2)/(1 + t^2)) - ((2t)/(1 + t^2))).(2dt)/(1 + t^2)`
= `int (1 + t^2)/(2 + 2t^2 + 1 - t^2 - 2t).(2dt)/(1 + t^2)`
= `2 int (1)/(t^2 - 2t + 3)dt`
= `2 int (1)/((t^2 - 2t + 1) + 2)dt`
= `2 int (1)/((t - 1)^2 + (sqrt(2))^2).dt`
= `2 xx (1)/sqrt(2)tan^-1 ((t - 1)/sqrt(2)) + c`
= `sqrt(2)tan^-1[(tan(x/2) - 1)/sqrt(2)] + c`.
APPEARS IN
संबंधित प्रश्न
Integrate the functions:
`sqrt(ax + b)`
Integrate the functions:
`(x^3 - 1)^(1/3) x^5`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`(1+ log x)^2/x`
Write a value of
Write a value of
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{9 + x^2} \text{ dx }\].
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
Integrate the following w.r.t. x : `(3x^3 - 2x + 5)/(xsqrt(x)`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
Integrate the following functions w.r.t. x : e3logx(x4 + 1)–1
Integrate the following function w.r.t. x:
`(10x^9 +10^x.log10)/(10^x + x^10)`
Evaluate the following : `int (1)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following:
`int (1)/sqrt((x - 3)(x + 2)).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Evaluate the following integrals : `int sqrt((9 - x)/x).dx`
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
`int logx/(log ex)^2*dx` = ______.
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int 1/(4x^2 - 20x + 17)` dx
`int sqrt(1 + "x"^2) "dx"` =
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
`int 2/(sqrtx - sqrt(x + 3))` dx = ________________
`int x^x (1 + logx) "d"x`
`int1/(4 + 3cos^2x)dx` = ______
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int[ tan (log x) + sec^2 (log x)] dx= ` ______
If `int x^3"e"^(x^2) "d"x = "e"^(x^2)/2 "f"(x) + "c"`, then f(x) = ______.
If `int(cosx - sinx)/sqrt(8 - sin2x)dx = asin^-1((sinx + cosx)/b) + c`. where c is a constant of integration, then the ordered pair (a, b) is equal to ______.
`int sqrt(x^2 - a^2)/x dx` = ______.
`int 1/(sinx.cos^2x)dx` = ______.
`int (logx)^2/x dx` = ______.
`int secx/(secx - tanx)dx` equals ______.
Evaluate the following.
`int 1/(x^2+4x-5) dx`
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate the following.
`intx sqrt(1 +x^2) dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
