Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `int (1)/(2sin 2x - 3)dx`
Advertisements
उत्तर
Let I = `int (1)/(2sin 2x - 3)dx`
Put tan x = t
∴ x = tan–1 t
∴ dx = `dt/(1 + t^2) and sin 2x = (2t)/(1 + t^2)`
∴ I = `int(1)/(2((2t)/(1 + t^2)) - 3).dt/(1 + t^2)`
= `int (1 + t^2)/(4t - 3 - 3t^2).dt/(1 + t^2)`
= `int (1)/(-3t^2 + 4t - 3)dt`
= `(1)/(3) int (1)/(t^2 - 4/3t + 1)dt`
= `-(1)/(3) int (1)/((t^2 - 4/3t + 4/9) - (4)/(9) + 1)dt`
= `-(1)/(3) int (1)/((t - 2/3)^2 + (sqrt(5)/3)^2)dt`
= `-(1)/(3) xx (1)/((sqrt(5)/3))tan^-1 ((t - 2/3)/(sqrt(5)/3)) + c`
= `-(1)/sqrt(5)tan^-1 ((3t - 2)/sqrt(5)) + c`
= `-(1)/sqrt(5)tan^-1((3tan x - 2)/(sqrt(5))) + c`.
APPEARS IN
संबंधित प्रश्न
Show that: `int1/(x^2sqrt(a^2+x^2))dx=-1/a^2(sqrt(a^2+x^2)/x)+c`
Evaluate :`intxlogxdx`
Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
sec2(7 – 4x)
Evaluate : `∫1/(3+2sinx+cosx)dx`
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of
Evaluate: \[\int\frac{x^3 - 1}{x^2} \text{ dx}\]
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Evaluate the following integrals : `int sinx/(1 + sinx)dx`
Evaluate the following integrals : `intsqrt(1 - cos 2x)dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t.x:
`(5 - 3x)(2 - 3x)^(-1/2)`
Integrate the following functions w.r.t. x : `3^(cos^2x) sin 2x`
Evaluate the following:
`int (1)/(25 - 9x^2)*dx`
Evaluate the following : `int (1)/(7 + 2x^2).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Evaluate the following : `int sinx/(sin 3x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Choose the correct option from the given alternatives :
`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =
Choose the correct options from the given alternatives :
`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =
Evaluate the following.
`int 1/("x" log "x")`dx
Fill in the Blank.
`int (5("x"^6 + 1))/("x"^2 + 1)` dx = x4 + ______ x3 + 5x + c
Evaluate `int (5"x" + 1)^(4/9)` dx
`int 1/(cos x - sin x)` dx = _______________
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int cot^2x "d"x`
`int cos^7 x "d"x`
`int(log(logx))/x "d"x`
`int (x^2 + 1)/(x^4 - x^2 + 1)`dx = ?
`int "dx"/((sin x + cos x)(2 cos x + sin x))` = ?
`int ("e"^x(x + 1))/(sin^2(x"e"^x)) "d"x` = ______.
`int(3x + 1)/(2x^2 - 2x + 3)dx` equals ______.
`int (x + sinx)/(1 + cosx)dx` is equal to ______.
Write `int cotx dx`.
Find `int (x + 2)/sqrt(x^2 - 4x - 5) dx`.
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Prove that:
`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate the following.
`int1/(x^2+4x-5) dx`
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4)dx`
Evaluate the following:
`int x^3/(sqrt(1 + x^4)) dx`
