Advertisements
Advertisements
Question
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
Solution
Let`I = int_(-pi//2)^(pi//2) sin^2 x dx`
`= 2 int_0^(pi//2) sin^2 x dx` ...(i) ...(∵ sin2 x is a function)
Then `I = 2 int_0^(pi//2) sin^2 (pi/2 - x) dx`
`= int_0^(pi//2) cos^2 x dx` ...(ii) `[because int_0^a f(x) = int_0^a f(a - x) dx]`
On adding equations (i) and (ii)
`2I = 2 int_0^(pi//2) (sin^2 x + cos^2 x) dx`
`2I = 2 int_0^(pi//2) 1 dx`
`=> 2I = 2 [x]_0^(pi//2)`
`=> 2I = 2 xx pi/2`
Hence, `I = pi/2`
APPEARS IN
RELATED QUESTIONS
Evaluate `int_(-2)^2x^2/(1+5^x)dx`
Evaluate: `int_(-a)^asqrt((a-x)/(a+x)) dx`
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_2^8 |x - 5| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}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))` .
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Evaluate : ∫ log (1 + x2) dx
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
`int_2^4 x/(x^2 + 1) "d"x` = ______
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
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` = ____________
`int_{pi/6}^{pi/3} sin^2x dx` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^9 1/(1 + sqrtx)` dx = ______
`int_0^1 "e"^(5logx) "d"x` = ______.
Find `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x)) "d"x`
`int_(-1)^1 (x^3 + |x| + 1)/(x^2 + 2|x| + 1) "d"x` is equal to ______.
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` 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 = ______.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Solve the following.
`int_1^3 x^2 logx dx`
`int_-9^9 x^3/(4-x^2) dx` =______
Evaluate: `int_-1^1 x^17.cos^4x dx`
Solve the following.
`int_2^3x/((x+2)(x+3))dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
