Advertisements
Advertisements
Question
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Solution
let I = `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Put x2 = t. to find constants A and B.
`"t"/("t"^2 + 5"t" + 6) = "t"/(("t" + 2)("t" + 3))`
`= "A"/("t" + 2) + "B"/"t + 3"`
∴ t = A (t + 3) + B (t + 2)
Putting t = -3 in equation (II).
-3 = B ( -1 ) ⇒ B = 3
Putting t = -2 in equation (II).
-2 = A (1) ⇒ A = -2
Substituting the values of A and replacing t by x2 in equation (I). we get
`"x"^2/("x"^4 + "5x"^2 + 6) = - 2/("x"^3 + 2) + 3/("x"^2 + 3)`
`therefore "I" = int [(-2)/("x"^2 + 2) + 3/("x"^2 + 3)] "dx"`
`= (-2) int "dx"/ ("x"^2 + 2) + 3 int "dx"/("x"^2 + 3)`
`= (-2) xx 1/sqrt2 "tan"^-1 ("x"/sqrt 2) + 3 xx 1/sqrt 3 "tan"^-1 ("x"/sqrt 3) + "c"`
`= -sqrt 2 "tan"^-1 ("x"/sqrt2) + sqrt 3 "tan"^-1 ("x"/sqrt 3) + "c"`
APPEARS IN
RELATED QUESTIONS
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_2^8 |x - 5| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate : ∫ log (1 + x2) dx
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
Evaluate `int_1^3 x^2*log x "d"x`
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_0^{pi/2} cos^2x dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^pi x*sin x*cos^4x "d"x` = ______.
Evaluate the following:
`int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
The value of the integral `int_0^sqrt(2)([sqrt(2 - x^2)] + 2x)dx` (where [.] denotes greatest integer function) is ______.
If `lim_("n"→∞)(int_(1/("n"+1))^(1/"n") tan^-1("n"x)"d"x)/(int_(1/("n"+1))^(1/"n") sin^-1("n"x)"d"x) = "p"/"q"`, (where p and q are coprime), then (p + q) is ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
