Advertisements
Advertisements
प्रश्न
Evaluate the following integral:
`int (3cosx)/(4sin^2x + 4sinx - 1).dx`
Advertisements
उत्तर
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
संबंधित प्रश्न
Integrate the functions:
`x/(e^(x^2))`
Integrate the functions:
sec2(7 – 4x)
Integrate the functions:
`(2cosx - 3sinx)/(6cos x + 4 sin x)`
Evaluate `int 1/(3+ 2 sinx + cosx) dx`
Write a value of
Write a value of
Write a value of\[\int\frac{1}{x \left( \log x \right)^n} \text { dx }\].
Integrate the following w.r.t. x : x3 + x2 – x + 1
Evaluate the following integrals : `int sqrt(1 + sin 2x) dx`
Evaluate the following integrals: `int sin 4x cos 3x dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t. x : `((sin^-1 x)^(3/2))/(sqrt(1 - x^2)`
Integrate the following functions w.r.t. x : `cosx/sin(x - a)`
Evaluate the following : `int (1)/sqrt(11 - 4x^2).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Integrate the following functions w.r.t. x : `int (1)/(4 - 5cosx).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following integrals:
`int (2x + 1)/(x^2 + 4x - 5).dx`
Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).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 :
`int f x^x (1 + log x)*dx`
`int logx/(log ex)^2*dx` = ______.
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Evaluate the following.
∫ (x + 1)(x + 2)7 (x + 3)dx
Evaluate the following.
`int 1/("x" log "x")`dx
Evaluate the following.
`int ((3"e")^"2t" + 5)/(4"e"^"2t" - 5)`dt
`int ("x + 2")/(2"x"^2 + 6"x" + 5)"dx" = "p" int (4"x" + 6)/(2"x"^2 + 6"x" + 5) "dx" + 1/2 int "dx"/(2"x"^2 + 6"x" + 5)`, then p = ?
To find the value of `int ((1 + log x) )/x dx` the proper substitution is ______.
Evaluate `int "x - 1"/sqrt("x + 4")` dx
`int e^x/x [x (log x)^2 + 2 log x]` dx = ______________
State whether the following statement is True or False:
`int3^(2x + 3) "d"x = (3^(2x + 3))/2 + "c"`
If `tan^-1x = 2tan^-1((1 - x)/(1 + x))`, then the value of x is ______
The value of `intsinx/(sinx - cosx)dx` equals ______.
`int x/sqrt(1 - 2x^4) dx` = ______.
(where c is a constant of integration)
Evaluate the following.
`int 1/(x^2+4x-5) dx`
Evaluate `int (1+x+x^2/(2!)) dx`
Evaluate:
`int(cos 2x)/sinx dx`
Evaluate:
`intsqrt(3 + 4x - 4x^2) dx`
Evaluate the following:
`int (1) / (x^2 + 4x - 5) dx`
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate `int (1 + "x" + "x"^2/(2!))`dx
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
If \[x=g(t)\] and \[dx=g'(t)dt\], which formula represents integration by substitution?
Which standard substitution is used for \[\sqrt{\mathrm{a}^2-x^2}\], \[\frac{1}{\sqrt{\mathrm{a}^2-x^2}}\], or \[\mathrm{a}^2-x^2\]?
For \[\int\frac{\sin x}{\sin(x+a)}\,dx\], which substitution gives \[dx=dt\]?
When applying substitution, what must always be rewritten in terms of the new variable?
