Advertisements
Advertisements
प्रश्न
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Advertisements
उत्तर
Let I = `int (1)/(2 + 3tanx).dx`
= `int(1)/(2 + 3(sinx/cosx)).dx`
= `int cosx/(2cosx + 3sinx).dx`
Put,
Numerator = `"A (Denominator) + B"[d/dx("Denominator")]`
∴ cos x = `"A"(2cosx + 3 sinx ) + "B"[d/dx(2cos x + 3 sin x)]`
= A(2 cos x + 3 sin x) + B(– 2 sin x + 3 cos x)
∴ cos x = (2A + 3B)cos x + (3A – 2B)sin x
Equating the coefficients of cos x sin x on both the sides, we get
2A 3B = 1 ...(1)
and
3A – 2B = 0 ...(2)
Multiplying equation (1) by 22 and equation (2) by 3, we get
4A +6B = 2
9A – 6B = 0
On adding, we get
13A = 2
∴ A = `(2)/(13)`
∴ from (2), 2B = 3A = `3(2/13) = (6)/(13)`
∴ B = `(3)/(13)`
∴ cos x = `(2)/(13)(2cosx + 3sinx) + (3)/(13)(-2sinx + 3cosx)`
∴ I = `int[(2/13(2cosx + 3sinx) + 3/13(-2 sinx + 3cosx))/(2cosx + 3sinx)].dx`
= `int[2/13 + (3/13(-2sinx + 3cosx))/(2cosx + 3sinx)].dx`
= `(2)/(13) 1 dx + (3)/(13) int (-2sinx + 3cosx)/(2cosx + 3sinx).dx`
= `(2)/(13)x + (3)/(13)log|2cos x + 3sinx| + c. ...[∵ int (f'(x))/f(x)dx = log|f(x)| + c]`
APPEARS IN
संबंधित प्रश्न
Prove that `int_a^bf(x)dx=f(a+b-x)dx.` Hence evaluate : `int_a^bf(x)/(f(x)+f(a-b-x))dx`
Find the particular solution of the differential equation x2dy = (2xy + y2) dx, given that y = 1 when x = 1.
Integrate the functions:
(4x + 2) `sqrt(x^2 + x +1)`
Integrate the functions:
`1/(x-sqrtx)`
Integrate the functions:
`1/(1 + cot x)`
Evaluate: `int (2y^2)/(y^2 + 4)dx`
Evaluate: `int (sec x)/(1 + cosec x) dx`
Write a value of
The value of \[\int\frac{1}{x + x \log x} dx\] is
Integrate the following w.r.t. x : `int x^2(1 - 2/x)^2 dx`
Integrate the following w.r.t. x:
`3 sec^2x - 4/x + 1/(xsqrt(x)) - 7`
Evaluate the following integrals : `int (cos2x)/(sin^2x.cos^2x)dx`
Evaluate the following integrals:
`int x/(x + 2).dx`
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Evaluate the following integrals: `int(x - 2)/sqrt(x + 5).dx`
Integrate the following functions w.r.t. x : `e^x.log (sin e^x)/tan(e^x)`
Integrate the following functions w.r.t.x:
`(2sinx cosx)/(3cos^2x + 4sin^2 x)`
Integrate the following functions w.r.t. x : `(1)/(x(x^3 - 1)`
Integrate the following functions w.r.t. x : `(1)/(x.logx.log(logx)`.
Integrate the following functions w.r.t. x:
`(1)/(sinx.cosx + 2cos^2x)`
Integrate the following functions w.r.t. x : `(3e^(2x) + 5)/(4e^(2x) - 5)`
Integrate the following functions w.r.t. x : tan 3x tan 2x tan x
Evaluate the following : `int (1)/sqrt(2x^2 - 5).dx`
Evaluate the following : `int (1)/sqrt(3x^2 + 5x + 7).dx`
Integrate the following functions w.r.t. x : `int (1)/(3 + 2sinx).dx`
Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`
Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`
Evaluate the following integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Evaluate the following : `int (logx)2.dx`
Evaluate `int (3"x"^3 - 2sqrt"x")/"x"` dx
Evaluate `int 1/(x (x - 1))` dx
Evaluate the following.
`int 1/("x"^2 + 4"x" - 5)` dx
Choose the correct alternative from the following.
The value of `int "dx"/sqrt"1 - x"` is
Evaluate: `int "e"^"x" (1 + "x")/(2 + "x")^2` dx
`int x/(x + 2) "d"x`
State whether the following statement is True or False:
`int"e"^(4x - 7) "d"x = ("e"^(4x - 7))/(-7) + "c"`
Evaluate `int"e"^x (1/x - 1/x^2) "d"x`
`int(1 - x)^(-2)` dx = `(1 - x)^(-1) + c`
If f′(x) = 4x3 − 3x2 + 2x + k, f(0) = -1 and f(1) = 4, find f(x)
Evaluate `int1/(x(x - 1))dx`
`int x^3 e^(x^2) dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3) dx`
Evaluate `int 1/(x(x-1))dx`
Evaluate `int(1+x+(x^2)/(2!))dx`
Evaluate `int (1 + x + x^2/(2!)) dx`
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 indefinite integrals, what should be done after integration in the new variable?
