Advertisements
Advertisements
Question
Integrate the following functions w.r.t. x : `(1)/(2 + 3tanx)`
Advertisements
Solution
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
RELATED QUESTIONS
Evaluate: `int sqrt(tanx)/(sinxcosx) dx`
Integrate the functions:
`1/(x(log x)^m), x > 0, m ne 1`
Integrate the functions:
`(sin^(-1) x)/(sqrt(1-x^2))`
Integrate the functions:
`1/(cos^2 x(1-tan x)^2`
Integrate the functions:
`(sin x)/(1+ cos x)^2`
Integrate the functions:
`1/(1 + cot x)`
Evaluate: `int 1/(x(x-1)) dx`
Write a value of\[\int\frac{1}{1 + 2 e^x} \text{ dx }\].
Write a value of\[\int\left( e^{x \log_e \text{ a}} + e^{a \log_e x} \right) dx\] .
Write a value of\[\int e^{ax} \sin\ bx\ dx\]
Write a value of\[\int e^{ax} \left\{ a f\left( x \right) + f'\left( x \right) \right\} dx\] .
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
Write a value of \[\int\frac{1 - \sin x}{\cos^2 x} \text{ dx }\]
Evaluate the following integrals : `int(5x + 2)/(3x - 4).dx`
Integrate the following functions w.r.t. x : cos7x
Evaluate the following : `(1)/(4x^2 - 20x + 17)`
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 integrals : `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`
Choose the correct options from the given alternatives :
`int (e^x(x - 1))/x^2*dx` =
Evaluate `int (3"x"^2 - 5)^2` dx
Evaluate the following.
`int (2"e"^"x" + 5)/(2"e"^"x" + 1)`dx
Evaluate the following.
`int 1/(4"x"^2 - 1)` dx
Evaluate the following.
`int 1/("a"^2 - "b"^2 "x"^2)` dx
Evaluate the following.
`int 1/(sqrt("x"^2 + 4"x"+ 29))` dx
Evaluate the following.
`int 1/(sqrt("x"^2 -8"x" - 20))` dx
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Evaluate: ∫ |x| dx if x < 0
`int sqrt(x^2 + 2x + 5)` dx = ______________
`int cos sqrtx` dx = _____________
If `int 1/(x + x^5)` dx = f(x) + c, then `int x^4/(x + x^5)`dx = ______
`int (1 + x)/(x + "e"^(-x)) "d"x`
`int1/(4 + 3cos^2x)dx` = ______
`int 1/(a^2 - x^2) dx = 1/(2a) xx` ______.
`int cos^3x dx` = ______.
Evaluate.
`int(5"x"^2 - 6"x" + 3)/(2"x" - 3) "dx"`
Evaluate the following.
`int 1/(x^2 + 4x - 5) dx`
Evaluate `int(1 + x + x^2/(2!))dx`
Evaluate the following.
`int x sqrt(1 + x^2) dx`
Evaluate:
`int 1/(1 + cosα . cosx)dx`
Evaluate:
`int sqrt((a - x)/x) dx`
If f ′(x) = 4x3 − 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x)
`int 1/(sin^2x cos^2x)dx` = ______.
Evaluate the following:
`int x^3/(sqrt(1+x^4))dx`
If f'(x) = 4x3 – 3x2 + 2x + k, f(0) = 1 and f(1) = 4, find f(x).
