Advertisements
Advertisements
Question
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
Options
4
2
1
0
Advertisements
Solution
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is 4.
Explanation:
`int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`
Taking LHS = `int_0^(2π) cos^2 x dx`
= `2int_0^π cos^2 x dx` ...[∵ cos2 x is an even function]
= `2 xx 2int_0^(π/2) cos^2 x dx` ...[∵ cos2 x is an even function]
= `4int_0^(π/2) cos^2 x dx`
On comparing both sides, we get
k = 4.
APPEARS IN
RELATED QUESTIONS
If `int_0^alpha(3x^2+2x+1)dx=14` then `alpha=`
(A) 1
(B) 2
(C) –1
(D) –2
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^2 xsqrt(2 -x)dx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.
`int_1^2 1/(2x + 3) dx` = ______
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
Evaluate `int_0^1 x(1 - x)^5 "d"x`
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`
`int_0^(pi/4) (sec^2 x)/((1 + tan x)(2 + tan x))`dx = ?
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_0^1 (1 - x/(1!) + x^2/(2!) - x^3/(3!) + ... "upto" ∞)` e2x dx = ?
`int_2^3 x/(x^2 - 1)` dx = ______
If `int_0^"a" sqrt("a - x"/x) "dx" = "K"/2`, then K = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
`int_0^pi sin^2x.cos^2x dx` = ______
`int_0^1 "e"^(5logx) "d"x` = ______.
`int_(-2)^2 |x cos pix| "d"x` is equal to ______.
Evaluate: `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tanx)`
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)?
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
Evaluate the following definite integral:
`int_1^3 log x dx`
Evaluate:
`int_0^1 |2x + 1|dx`
Evaluate the following integral:
`int_-9^9 x^3 / (4 - x^2) dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
