Advertisements
Advertisements
प्रश्न
Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Advertisements
उत्तर
Let I = `int x^2/((x^2 + 1)(3x^2 + 4))dx`
Put t = x2
`t/((t + 1)(3t + 4)) = A/(t + 1) + B/(3t + 4)`
t = A(3t + 4) + B(t + 1)
t = (3A + B)t + (4A + B)
On comparing both sides, we get
3A + B = 1 and 4A + B = 0
∴ I = `int (-1)/(x^2 + 1)dx + int (-4)/(3x^2 + 4)dx`
= `-int 1/(x^2 + 1)dx - 4int 1/(3x^2 + 4)dx`
= `-int 1/(x^2 + 1^2)dx - 4int 1/((sqrt(3)x)^2 + 2^2)dx`
= `(-1)/1 tan^-1 (x/2) - 4/(2sqrt(3)) tan^-1 ((sqrt(3)x)/2) + C`
= `-tan^-1x - 2/sqrt(3) tan^-1 ((sqrt(3)x)/2) + C`
संबंधित प्रश्न
Evaluate : `int x^2/((x^2+2)(2x^2+1))dx`
Integrate the rational function:
`1/(x^2 - 9)`
Integrate the rational function:
`(1 - x^2)/(x(1-2x))`
Integrate the rational function:
`2/((1-x)(1+x^2))`
Integrate the rational function:
`(2x)/((x^2 + 1)(x^2 + 3))`
Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`
Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`
Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`
Integrate the following w.r.t. x : `(1)/(sin2x + cosx)`
Integrate the following w.r.t. x: `(2x^2 - 1)/(x^4 + 9x^2 + 20)`
Integrate the following with respect to the respective variable : `cot^-1 ((1 + sinx)/cosx)`
Integrate the following w.r.t.x : `sqrt(tanx)/(sinx*cosx)`
Evaluate: `int (2"x" + 1)/(("x + 1")("x - 2"))` dx
Evaluate: `int 1/("x"("x"^"n" + 1))` dx
Evaluate: `int (2"x"^3 - 3"x"^2 - 9"x" + 1)/("2x"^2 - "x" - 10)` dx
Evaluate: `int (1 + log "x")/("x"(3 + log "x")(2 + 3 log "x"))` dx
`int 1/(4x^2 - 20x + 17) "d"x`
`int 1/(2 + cosx - sinx) "d"x`
Choose the correct alternative:
`int sqrt(1 + x) "d"x` =
If f'(x) = `1/x + x` and f(1) = `5/2`, then f(x) = log x + `x^2/2` + ______ + c
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
`int x/((x - 1)^2 (x + 2)) "d"x`
If `int(sin2x)/(sin5x sin3x)dx = 1/3log|sin 3x| - 1/5log|f(x)| + c`, then f(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 (x^2"d"x)/(x^4 - x^2 - 12)`
Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`
Value of ∫ `(x^2 + 1)/((x − 1)(x − 2))`dx is ______.
