Advertisements
Advertisements
Question
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Advertisements
Solution
Let I = `int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Put sin x = t
∴ cosx dx = dt
∴ I = `int 3/(4t^2 + 4t - 1)dt`
I = `3/4 int 1/(t^2 + t - 1/4)dt`
I = `3/4 int 1/((t^2 + t + 1/4) - 1/4 - 1/4)dt`
I = `3/4 int 1/ ((t + 1/2)^2 - 1/2)dt`
I = `3/4 int 1/sqrt((t + 1/2)^2 - (1/sqrt2)^2)dt`
`[∵ int 1/(x^2 - a^2)dx = 1/(2a) log |(x - a)/(x + a)| + c]`
I = `3/4 xx 1/(2(1/sqrt2)) log |(t + 1/2 - 1/sqrt2)/(t + 1/2 + 1/sqrt2)| + c`
I = `3/(4sqrt2) log |(2sqrt2t + (2sqrt2)/2 - (2sqrt2)/sqrt2)/(2sqrt2t + (2sqrt2)/2 - (2sqrt2)/sqrt2)| + c`
I = `3/(4sqrt2) log |(2sqrt2t + sqrt2 - 2)/(2sqrt2t +sqrt2 + 2)| + c`
I = `3/(4sqrt2) log |(2sqrt2sin + sqrt2 - 2)/(2sqrt2sin +sqrt2 + 2)| + c`
APPEARS IN
RELATED QUESTIONS
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`sin x/(1+ cos x)`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of
Write a value of
Write a value of\[\int\frac{\sin 2x}{a^2 \sin^2 x + b^2 \cos^2 x} \text{ dx }\]
Write a value of
Integrate the following functions w.r.t. x : `(x^2 + 2)/((x^2 + 1)).a^(x + tan^-1x)`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(1)/(sqrt(x) + sqrt(x^3)`
Integrate the following functions w.r.t. x : `x^2/sqrt(9 - x^6)`
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Evaluate the following : `int sqrt((9 + x)/(9 - x)).dx`
Evaluate the following : `int (1)/(cos2x + 3sin^2x).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 - 2cos 2x).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate `int 1/("x" ("x" - 1))` dx
If f'(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
Evaluate the following.
`int "x" sqrt(1 + "x"^2)` dx
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
`int 1/(cos x - sin x)` dx = _______________
`int "e"^x[((x + 3))/((x + 4)^2)] "d"x`
`int(log(logx))/x "d"x`
`int(5x + 2)/(3x - 4) dx` = ______
If I = `int (sin2x)/(3x + 4cosx)^3 "d"x`, then I is equal to ______.
`int ("d"x)/(x(x^4 + 1))` = ______.
The value of `intsinx/(sinx - cosx)dx` equals ______.
`int (logx)^2/x dx` = ______.
Evaluate `int(1 + x + x^2/(2!) )dx`
Evaluate `int(1+ x + x^2/(2!)) dx`
Evaluate:
`int(sqrt(tanx) + sqrt(cotx))dx`
Evaluate the following.
`intxsqrt(1+x^2)dx`
Evaluate `int(1+x+x^2/(2!))dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int1/(x(x-1))dx`
Evaluate the following.
`int1/(x^2+4x-5)dx`
Evaluate `int 1/(x(x-1)) dx`
