Advertisements
Advertisements
Question
Evaluate: `∫8/((x+2)(x^2+4))dx`
Advertisements
Solution
Let `I=∫8/((x+2)(x^2+4))dx`
Let `8/((x+2)(x^2+4))=A/(x+2)+(Bx+C)/(x^2+4)`
`8=A(x^2+4)+(Bx+C)(x+2)`
`8=A(x^2+4)+Bx^2+2Bx+Cx+2C`
`8=(A+B)x^2+(2B+c)x+(4A+2C)`
Comparing the coefficients of x2 , x and the constant term, we get
A + B = 0, 2B + C = 0 and 4A + 2C = 8
On solving these equations, we get
A = 1, B = –1, C = 2
`8/((x+2)(x^2+4))=1/(x+2)+(-x+2)/(x^2+4)`
`I=int[1/(x+2)+(-x+2)/(x^2+4)]dx`
`=int1/(x+2)dx-1/2int(2x)/(x^2+4)dx+2int1/(x^2+2^2)dx`
`=log|x+2|-1/2log|x^2+4|+tan^-1(x/2)+c`
`=log|(x+2)/sqrt(x^2+4)|+tan^-1(x/2)+c`
APPEARS IN
RELATED QUESTIONS
Find : `int x^2/(x^4+x^2-2) dx`
Integrate the rational function:
`(3x - 1)/((x - 1)(x - 2)(x - 3))`
Integrate the rational function:
`(5x)/((x + 1)(x^2 - 4))`
Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`
Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
Integrate the following w.r.t. x:
`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`
Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x: `(1)/(sinx + sin2x)`
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)`
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
`int x^7/(1 + x^4)^2 "d"x`
`int x^2sqrt("a"^2 - x^6) "d"x`
`int sqrt(4^x(4^x + 4)) "d"x`
`int ((x^2 + 2))/(x^2 + 1) "a"^(x + tan^(-1_x)) "d"x`
`int sec^3x "d"x`
`int sec^2x sqrt(tan^2x + tanx - 7) "d"x`
`int "e"^(sin^(-1_x))[(x + sqrt(1 - x^2))/sqrt(1 - x^2)] "d"x`
`int (x^2 + x -1)/(x^2 + x - 6) "d"x`
`int 1/(sinx(3 + 2cosx)) "d"x`
`int 1/(4x^2 - 20x + 17) "d"x`
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_"0"^pi (x"d"x)/(1 + sin x)`
Evaluate the following:
`int (2x - 1)/((x - 1)(x + 2)(x - 3)) "d"x`
If f(x) = `int(3x - 1)x(x + 1)(18x^11 + 15x^10 - 10x^9)^(1/6)dx`, where f(0) = 0, is in the form of `((18x^α + 15x^β - 10x^γ)^δ)/θ`, then (3α + 4β + 5γ + 6δ + 7θ) is ______. (Where δ is a rational number in its simplest form)
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
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
Integration by partial fractions is a method used to integrate which functions?
A proper rational function can be expressed as a sum of simpler rational functions called what?
Which partial-fraction decomposition is appropriate for \[\frac{\mathrm{p}x+\mathrm{q}}{(x-\mathrm{a})(x-\mathrm{b})}\]?
Which partial form corresponds to \[\frac{\mathrm{p}x^{2}+\mathrm{q}x+\mathrm{r}}{(x-\mathrm{a})(x-\mathrm{b})(x-\mathrm{c})}\]?
Which factorisation is correct for \[x^{2}-5x+6\]?
What must be included for a repeated linear factor?
