Advertisements
Advertisements
Question
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
Advertisements
Solution
We have I = `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
= `int_0^(pi/4) sqrt((sinx + cosx)^2) "d"x`
= `int_0^(pi/4) (sinx + cosx) "d"x`
= `(-cosx + sinx)_0^(pi/4)`
I = 1.
APPEARS IN
RELATED QUESTIONS
Evaluate : `intlogx/(1+logx)^2dx`
Evaluate : `intsec^nxtanxdx`
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`
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 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`
Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx` if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
`∫_4^9 1/sqrtxdx=`_____
(A) 1
(B) –2
(C) 2
(D) –1
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
Evaluate`int (1)/(x(3+log x))dx`
Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx` and hence evaluate `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Evaluate : ∫ log (1 + x2) dx
`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x)) dx` = ______.
`int_(-7)^7 x^3/(x^2 + 7) "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 (cos x + x sin x)/(x(x + cos x))`dx = ?
The value of `int_-3^3 ("a"x^5 + "b"x^3 + "c"x + "k")"dx"`, where a, b, c, k are constants, depends only on ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
f(x) = `{:{(x^3/k; 0 ≤ x ≤ 2), (0; "otherwise"):}` is a p.d.f. of X. The value of k is ______
`int_0^{pi/2} cos^2x dx` = ______
`int_3^9 x^3/((12 - x)^3 + x^3)` dx = ______
`int_0^1 log(1/x - 1) "dx"` = ______.
`int_0^pi x*sin x*cos^4x "d"x` = ______.
`int_(-1)^1 (x + x^3)/(9 - x^2) "d"x` = ______.
Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
Solve the following.
`int_1^3 x^2 logx dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Evaluate:
`int_0^6 |x + 3|dx`
