Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(sinx + 2cosx)/(3sinx + 4cosx)`
Advertisements
उत्तर
Let I = `int (sinx + 2cosx)/(3sinx + 4cosx).dx`
Put,
Numberator = `"A (Denominator) + B"[d/dx("Denominator")]`
∴ sinx + 2cosx = `"A"(3sinx + 4cosx) + "B"[d/dx(3sinx + 4cosx)]`
= A(3 sin x + 4 cos x) + B(3 cos x – 4 sin x)
∴ sin x + 2 cos x = (3A – 4B)sin x + (4A + 3B)cos x
Equaliting the coefficients of sin x and cos x on both the sides, we get
3A – 4B = 1 ...(1)
and
4A + 3B = 2 ...(2)
Multiplying equation (1) by 3 and equation (2) by 4, we get
9A – 12B = 3
16A + 12B = 8
On adding, we get
25A = 11
∴ A = `(11)/(25)`
∴ from (2), `4(11/25) + 3"B"` = 2
∴ 3B = `2 - (44)/(25) = (6)/(25)`
∴ B = `(2)/(25)`
∴ `sinx + 2cos x = (11)/(25)(3sinx + 4cosx) + (2)/(25)(3cosx - 4sinx)`
∴ I = `int[(11/25(3sinx + 4cosx) + 2/25(3cosx - 4sinx))/(3sinx + 4cosx)].dx`
= `int[11/25 + (2/25(3cosx - 4sinx))/((3sinx + 4cosx))].dx`
= `(11)/(25) int1 dx + 2/25 int(3cosx - 4sin x)/(3sin x + 4cosx).dx`
= `(11)/(25)x + (2)/(25)log|3 sin x + 4 cos x| + c. ...[∵ int (f'(x))/(f'(x))dx = log|f(x)| + c]`
APPEARS IN
संबंधित प्रश्न
Find : `int((2x-5)e^(2x))/(2x-3)^3dx`
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`e^(tan^(-1)x)/(1+x^2)`
Integrate the functions:
`(e^(2x) - e^(-2x))/(e^(2x) + e^(-2x))`
Integrate the functions:
tan2(2x – 3)
Integrate the functions:
`cos sqrt(x)/sqrtx`
Integrate the functions:
`cos x /(sqrt(1+sinx))`
Integrate the functions:
`((x+1)(x + logx)^2)/x`
Integrate the functions:
`(x^3 sin(tan^(-1) x^4))/(1 + x^8)`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
The value of \[\int\frac{1}{x + x \log x} dx\] is
\[\int\frac{\sin x + 2 \cos x}{2 \sin x + \cos x} \text{ dx }\]
`int "dx"/(9"x"^2 + 1)= ______. `
Evaluate the following integrals : `int sin x/cos^2x dx`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Integrate the following functions w.r.t. x : sin4x.cos3x
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)/sqrt(3x^2 - 8).dx`
Evaluate the following : `int sqrt((10 + x)/(10 - x)).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Evaluate the following : `int (logx)2.dx`
Choose the correct options from the given alternatives :
`int sqrt(cotx)/(sinx*cosx)*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 w.r.t.x: `(3x + 1)/sqrt(-2x^2 + x + 3)`
If f '(x) = `"x"^2/2 - "kx" + 1`, f(0) = 2 and f(3) = 5, find f(x).
Evaluate the following.
`int "x"^3/(16"x"^8 - 25)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
Evaluate: `int "e"^sqrt"x"` dx
`int x^2/sqrt(1 - x^6)` dx = ________________
`int (sin4x)/(cos 2x) "d"x`
`int cot^2x "d"x`
`int sqrt(("e"^(3x) - "e"^(2x))/("e"^x + 1)) "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"`
Evaluate `int(3x^2 - 5)^2 "d"x`
If f(x) = 3x + 6, g(x) = 4x + k and fog (x) = gof (x) then k = ______.
Evaluate:
`int sqrt((a - x)/x) dx`
Evaluate:
`int(cos 2x)/sinx dx`
`int (cos4x)/(sin2x + cos2x)dx` = ______.
Evaluate the following.
`int1/(x^2 + 4x - 5) dx`
Evaluate the following.
`int1/(x^2 + 4x - 5)dx`
Evaluate the following.
`intx^3/sqrt(1 + x^4) dx`
Which substitution is appropriate for \[\sqrt{\frac{\mathrm{a}-x}{\mathrm{a}+x}}\] or \[\sqrt{\frac{\mathrm{a}+x}{\mathrm{a}-x}}\]?
Which expression results after simplifying \[\int\frac{\sin(t-a)}{\sin t}\,dt\]?
