Advertisements
Advertisements
Question
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)
Advertisements
Solution
Let I = `int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2`
Dividing the numerator and denominator by cos4x, we have
I = `int_0^(pi/2) (sec^4x)/(("a"^2 cos^2x)/(cos^2x) + ("b"^2 sin^2x)/cos^2x)^2 "d"x`
= `int_0^(pi/2) (sec^2x * sec^2x)/("a"^2 + "b"^2 tan^2 x)^2 "d"x`
= `int_0^(pi/2) ((1 + tan^2x) sec^2x)/("a"^2 + "b"^2 tan^2 x)^2 "d"x`
Put tan x = t
⇒ sec2x dx = dt
Changing the limits, we get
When x = 0
t = tan 0 = 0
When x = `pi/2`
t = `tan pi/2 = oo`
∴ I = `int_0^oo (1 + "t"^2)/("a"^2 + "b"^2"t"^2)^2 "dt"`
Put t2 = u only for the purpose of partial fraction
∴ `(1 +"u")/("a"^2 + "b"^2"u")^2 = "A"/(("a"^2 + "b"^2"u")) + "B"/("a"^2 + "b"^2"u")^2`
1 + u = A(a2 + b2u) + B
Comparing the coefficients of like terms, we get
a2A + B = 1 and b2A = 1
⇒ A = `1/"b"^2`
Now `"a"^2 * 1/"b"^2 + "B"` = 1
⇒ B = `1 - "a"^2/"b"^2`
= `("b"^2 - "a"^2)/"b"^2`
∴ I = `int_0^oo (1 + "t"^2)/("a"^2 + "b"^2"t"^2)^2`
= `1/"b"^2 int_0^oo "dt"/("a"^2 + "b"^2"t"^2) + ("b"^2 - "a"^2)/"b"^2 int_0^oo "dt"/("a"^2 + "b"^2"t"^2)^2`
= `1/"b"^2 int_0^oo "dt"/("b"^2("a"^2/"b"^2 + "t"^2)) + ("b"^2 - "a"^2)/"b"^2 int_0^oo "dt"/("a"^2 + "b"^2"t"^2)^2`
= `1/"ab"^3 [tan^-1 "t"/("a"/"b")]_0^oo + ("b"^2 - "a"^2)/"b"^2 (pi/4 * 1/("a"^3"b"))`
= `1/"ab"^3 [tan^-1 oo - tan 0] + ("b"^2 - "a"^2)/"b"^2 (pi/(4"a"^3"b"))`
= `1/"ab"^3 * pi/2 + pi/4 * ("b"^2 - "a"^2)/("a"^2"b"^3)`
= `pi/(2"ab"^3) + pi/4 * ("b"^2 - "a"^2)/("a"^3"b"^3)`
= `pi [(2"a"^2 + "b"^2 - "a"^2)/(4"a"^3"b"^3)]`
= `pi/4 (("a"^2 + "b"^2)/("a"^3"b"^3))`
APPEARS IN
RELATED QUESTIONS
Evaluate: `int_(-a)^asqrt((a-x)/(a+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^(2x) cos^5 xdx`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
By completing the following activity, Evaluate `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x)) "d"x`.
Solution: Let I = `int_2^5 (sqrt(x))/(sqrt(x) + sqrt(7 - x)) "d"x` ......(i)
Using the property, `int_"a"^"b" "f"(x) "d"x = int_"a"^"b" "f"("a" + "b" - x) "d"x`, we get
I = `int_2^5 ("( )")/(sqrt(7 - x) + "( )") "d"x` ......(ii)
Adding equations (i) and (ii), we get
2I = `int_2^5 (sqrt(x))/(sqrt(x) - sqrt(7 - x)) "d"x + ( ) "d"x`
2I = `int_2^5 (("( )" + "( )")/("( )" + "( )")) "d"x`
2I = `square`
∴ I = `square`
The c.d.f, F(x) associated with p.d.f. f(x) = 3(1- 2x2). If 0 < x < 1 is k`(x - (2x^3)/"k")`, then value of k is ______.
`int_0^{pi/2} log(tanx)dx` = ______
`int_0^{pi/2} xsinx dx` = ______
If `int_0^"a" sqrt("a - x"/x) "dx" = "K"/2`, then K = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
Let f be a real valued continuous function on [0, 1] and f(x) = `x + int_0^1 (x - t)f(t)dt`. Then, which of the following points (x, y) lies on the curve y = f(x)?
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
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 ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
`int_1^2 x logx dx`= ______
Evaluate:
`int_0^1 |2x + 1|dx`
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
`int_0^(pi/4) (cos^2 x)/(cos^2 x + 4 sin^2 x) dx` =
