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:
`1/(x^2 - 9)`
Integrate the rational function:
`x/((x-1)(x- 2)(x - 3))`
Integrate the rational function:
`(2x)/(x^2 + 3x + 2)`
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 (dx)/(x(x^2 + 1))` equals:
Evaluate : `∫(x+1)/((x+2)(x+3))dx`
Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t. x : `(1)/(sinx*(3 + 2cosx)`
Integrate the following w.r.t. x : `(2log x + 3)/(x(3 log x + 2)[(logx)^2 + 1]`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `(cos 7x - cos8x)/(1 + 2 cos 5x)`
Integrate the following w.r.t.x : `(1)/(2cosx + 3sinx)`
Integrate the following w.r.t.x:
`x^2/((x - 1)(3x - 1)(3x - 2)`
Integrate the following w.r.t.x : `(1)/(sinx + sin2x)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int ("x"^2 + "x" - 1)/("x"^2 + "x" - 6)` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
`int "dx"/(("x" - 8)("x" + 7))`=
`int (2x - 7)/sqrt(4x- 1) dx`
`int sqrt(4^x(4^x + 4)) "d"x`
`int 1/(x(x^3 - 1)) "d"x`
If f'(x) = `x - 3/x^3`, f(1) = `11/2` find f(x)
`int sqrt((9 + x)/(9 - x)) "d"x`
`int (sinx)/(sin3x) "d"x`
`int sec^3x "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int (3x + 4)/sqrt(2x^2 + 2x + 1) "d"x`
`int x^3tan^(-1)x "d"x`
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
`int ("d"x)/(x^3 - 1)`
`int xcos^3x "d"x`
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`
`int x/((x - 1)^2 (x + 2)) "d"x`
`int (3"e"^(2"t") + 5)/(4"e"^(2"t") - 5) "dt"`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(x) = ______
If `intsqrt((x - 7)/(x - 9)) dx = Asqrt(x^2 - 16x + 63) + log|x - 8 + sqrt(x^2 - 16x + 63)| + c`, then A = ______
Verify the following using the concept of integration as an antiderivative
`int (x^3"d"x)/(x + 1) = x - x^2/2 + x^3/3 - log|x + 1| + "C"`
Evaluate the following:
`int x^2/(1 - x^4) "d"x` put x2 = t
Evaluate the following:
`int (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate the following:
`int_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int "e"^(-3x) cos^3x "d"x`
If `int "dx"/((x + 2)(x^2 + 1)) = "a"log|1 + x^2| + "b" tan^-1x + 1/5 log|x + 2| + "C"`, then ______.
If `intsqrt((x - 5)/(x - 7))dx = Asqrt(x^2 - 12x + 35) + log|x| - 6 + sqrt(x^2 - 12x + 35) + C|`, then A = ______.
If \[\int\frac{2x+3}{(x-1)(x^{2}+1)}\mathrm{d}x\] = \[=\log_{e}\left\{(x-1)^{\frac{5}{2}}\left(x^{2}+1\right)^{a}\right\}-\frac{1}{2}\tan^{-1}x+\mathrm{A}\] where A is an arbitrary constant, then the value of a is
