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
Find: `int(x+3)sqrt(3-4x-x^2dx)`
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Evaluate :
`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`
Integrate the functions:
`1/(x + x log x)`
Integrate the functions:
`(1+ log x)^2/x`
Write a value of
Write a value of\[\int e^x \left( \frac{1}{x} - \frac{1}{x^2} \right) dx\] .
Integrate the following w.r.t. x:
`2x^3 - 5x + 3/x + 4/x^5`
Evaluate the following integral:
`int(4x + 3)/(2x + 1).dx`
Integrate the following functions w.r.t. x : `(logx)^n/x`
Integrate the following functions w.r.t. x : `(e^(2x) + 1)/(e^(2x) - 1)`
Integrate the following functions w.r.t. x : `sqrt(tanx)/(sinx.cosx)`
Integrate the following functions w.r.t. x : `(2x + 1)sqrt(x + 2)`
Integrate the following functions w.r.t. x : `sin(x - a)/cos(x + b)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sin x - cosx)dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2 sin2x + 4cos 2x).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).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` =
Integrate the following with respect to the respective variable : `(x - 2)^2sqrt(x)`
Integrate the following w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
Evaluate the following.
`int ("e"^"x" + "e"^(- "x"))^2 ("e"^"x" - "e"^(-"x"))`dx
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Evaluate the following.
`int 1/(sqrt(3"x"^2 - 5))` dx
`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 = ?
Evaluate `int 1/((2"x" + 3))` dx
`int sqrt(1 + sin2x) dx`
`int (cos2x)/(sin^2x) "d"x`
State whether the following statement is True or False:
`int sqrt(1 + x^2) *x "d"x = 1/3(1 + x^2)^(3/2) + "c"`
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 ______.
The value of `int (sinx + cosx)/sqrt(1 - sin2x) dx` is equal to ______.
`int dx/(2 + cos x)` = ______.
(where C is a constant of integration)
Evaluated the following
`int x^3/ sqrt (1 + x^4 )dx`
Evaluate `int (1+x+x^2/(2!))dx`
Evaluate the following.
`int x^3/(sqrt(1 + x^4))dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate the following
`int x^3/sqrt(1+x^4) dx`
Evaluate:
`int(5x^2-6x+3)/(2x-3)dx`
Evaluate the following.
`int "x"^3/sqrt(1 + "x"^4)` dx
Evaluate the following.
`int 1/ (x^2 + 4x - 5) dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
For \[t=\cos x\], what is \[dt\]?
For \[\int\frac{\sin x}{\sin(x+a)}\,dx\], which substitution gives \[dx=dt\]?
Integration by substitution is the reverse process of which rule?
When applying substitution, what must always be rewritten in terms of the new variable?
