Advertisements
Advertisements
Question
`int ("d"x)/(2 + 3tanx)`
Advertisements
Solution
Let I = `int 1/(2 + 3tanx) "d"x`
= `int 1/(2 + 3(sinx/cosx)) "d"x`
= `int cosx/(2cosx + 3sinx) "d"x`
Let cos x = `"A"(2cosx + 3 sinx) + "B""d"/("d"x) (2cosx + 3sinx)`
= A(2cos x + 3sin x) + B(−2sin x + 3cos x)
∴ cos x + 0⋅sinx = cosx (2A + 3B) + sinx (3A − 2B)
By equating the coefficients on both sides, we get
2A + 3B = 1 and 3A − 2B = 0
Solving these equations, we get
A = `2/13` and B = `3/13`
∴ cos x = `2/13 (2 cos x + 3 sin x) + 3/13 (-2 sin x + 3 cos x)`
∴ I = `int (2/13(2cos x + 3sin x) + 3/13(-2 sinx + 3cos x))/(2cosx + 3sin x) "d"x`
∴ I = `2/13 int "d"x + 3/13 int (-2sinx + 3cosx)/(2cosx + 3sinx) "d"x`
∴ I = `2/13x + 3/13 log |2cos + 3sinx| + "c"` ........`[∵ int ("f'"(x))/("f"(x)) "d"x = log |"f"(x)| + "c"]`
APPEARS IN
RELATED QUESTIONS
Find: `I=intdx/(sinx+sin2x)`
Integrate the rational function:
`x/((x + 1)(x+ 2))`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(cos x)/((1-sinx)(2 - sin x))` [Hint: Put sin x = t]
Integrate the rational function:
`1/(e^x -1)`[Hint: Put ex = t]
`int (xdx)/((x - 1)(x - 2))` equals:
`int (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`
Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`
Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `((3sin - 2)*cosx)/(5 - 4sin x - cos^2x)`
Integrate the following w.r.t. x : `(5*e^x)/((e^x + 1)(e^(2x) + 9)`
Choose the correct options from the given alternatives :
If `int tan^3x*sec^3x*dx = (1/m)sec^mx - (1/n)sec^n x + c, "then" (m, n)` =
Integrate the following with respect to the respective variable : `(6x + 5)^(3/2)`
Integrate the following w.r.t. x: `(x^2 + 3)/((x^2 - 1)(x^2 - 2)`
Integrate the following w.r.t.x : `x^2/sqrt(1 - x^6)`
Integrate the following w.r.t.x : `(1)/((1 - cos4x)(3 - cot2x)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Evaluate:
`int (2x + 1)/(x(x - 1)(x - 4)) dx`.
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int 1/("x"("x"^5 + 1))` dx
State whether the following statement is True or False.
If `int (("x - 1") "dx")/(("x + 1")("x - 2"))` = A log |x + 1| + B log |x - 2| + c, then A + B = 1.
`int "e"^(3logx) (x^4 + 1)^(-1) "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
`int sec^3x "d"x`
`int sin(logx) "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int ("d"x)/(x^3 - 1)`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
Choose the correct alternative:
`int (x + 2)/(2x^2 + 6x + 5) "d"x = "p"int (4x + 6)/(2x^2 + 6x + 5) "d"x + 1/2 int 1/(2x^2 + 6x + 5)"d"x`, then p = ?
State whether the following statement is True or False:
For `int (x - 1)/(x + 1)^3 "e"^x"d"x` = ex f(x) + c, f(x) = (x + 1)2
Evaluate `int (2"e"^x + 5)/(2"e"^x + 1) "d"x`
Evaluate `int x^2"e"^(4x) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
Evaluate the following:
`int (x^2 "d"x)/((x^2 + "a"^2)(x^2 + "b"^2))`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
The numerator of a fraction is 4 less than its denominator. If the numerator is decreased by 2 and the denominator is increased by 1, the denominator becomes eight times the numerator. Find the fraction.
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
Evaluate:
`int 2/((1 - x)(1 + x^2))dx`
Evaluate:
`int x/((x + 2)(x - 1)^2)dx`
Evaluate.
`int (5x^2 - 6x + 3)/(2x - 3)dx`
Evaluate:
`int(2x^3 - 1)/(x^4 + x)dx`
