Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) cos^2 x dx`
Advertisements
उत्तर
Let `I = int_0^(pi/2) cos^2 x dx` ....(i)
and `I = int_0^(pi/2) cos^2 (pi/2 - x) dx`
`= int_0^(pi/2) sin^2 x dx` ....(ii) `[∵ int_0^a f (x) dx = int_0^a f (a - x) dx]`
Adding (i) and (ii), we get
`2 I = int_0^(pi/2) cos^2 x dx + int_0^(pi/2) sin^2 x dx`
`= int_0^(pi/2) (sin^2 x + cos^2 x) dx`
`= int_0^(pi/2) dx = [x]_0^(pi/2)`
`= pi/2`
⇒ `I = pi/4.`
APPEARS IN
संबंधित प्रश्न
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
Evaluate :`int_0^pi(xsinx)/(1+sinx)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_0^(pi/2) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) 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"`
`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
Evaluate `int_1^3 x^2*log x "d"x`
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^4 1/(1 + sqrtx)`dx = ______.
`int_-9^9 x^3/(4 - x^2)` dx = ______
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
`int_0^1 "e"^(5logx) "d"x` = ______.
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
`int_(-2)^2 |x cos pix| "d"x` is equal to ______.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) 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 integral `int_0^2||x - 1| -x|dx` is equal to ______.
`int_0^1|3x - 1|dx` equals ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Solve the following.
`int_1^3 x^2 logx dx`
`int_1^2 x logx dx`= ______
Evaluate the following integral:
`int_0^1 x(1 - x)^5 dx`
Evaluate the following integrals:
`int_-9^9 x^3/(4 - x^3 ) dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
