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_(-a)^asqrt((a-x)/(a+x)) 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_(pi/2)^(pi/2) sin^7 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
Using properties of definite integrals, evaluate
`int_0^(π/2) sqrt(sin x )/ (sqrtsin x + sqrtcos x)dx`
Find : `int_ (2"x"+1)/(("x"^2+1)("x"^2+4))d"x"`.
`int_1^2 1/(2x + 3) dx` = ______
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
Evaluate `int_1^3 x^2*log x "d"x`
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_0^pi x sin^2x dx` = ______
`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 the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
`int_0^1|3x - 1|dx` equals ______.
`int_0^(pi/4) (sec^2x)/((1 + tanx)(2 + tanx))dx` equals ______.
`int_0^(π/2)((root(n)(secx))/(root(n)(secx + root(n)("cosec" x))))dx` is equal to ______.
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`.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate: `int_0^π x/(1 + sinx)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:
`int_0^6 |x + 3|dx`
`∫_0^(π/2) (sqrttan x + sqrtcot x)dx` = ______.
Which expression equals \[\int_{a}^{b} f(x)\,dx\] by \[P_3\] : The "King's Rule"?
For an Even Function, which condition is satisfied?
Which expression is equal to \[\frac{1}{1+\sqrt{\tan x}}\] in the evaluation of \[\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{\tan x}}\]?
