Advertisements
Advertisements
प्रश्न
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Advertisements
उत्तर
`int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx = int_(-π//4)^(π//4) (2 cos^2 x - 1)/(2 cos^2 x)dx`
= `1/2 . 2 int_0^(π//4) (2 - sec^2 x)dx` ...[even function]
= `1/2 . 2[2x - tan x]_0^(π//4)`
= `π/2 - 1`
APPEARS IN
संबंधित प्रश्न
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_0^(pi/2) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
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_0^1 ((x^2 - 2)/(x^2 + 1))`dx = ?
`int_(pi/18)^((4pi)/9) (2 sqrt(sin x))/(sqrt (sin x) + sqrt(cos x))` dx = ?
`int_0^pi sin^2x.cos^2x dx` = ______
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
`int_0^(2"a") "f"(x) "d"x = 2int_0^"a" "f"(x) "d"x`, if f(2a – x) = ______.
`int_0^(2"a") "f"("x") "dx" = int_0^"a" "f"("x") "dx" + int_0^"a" "f"("k" - "x") "dx"`, then the value of k is:
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
`int_a^b f(x)dx = int_a^b f(x - a - b)dx`.
`int_0^5 cos(π(x - [x/2]))dx` where [t] denotes greatest integer less than or equal to t, is equal to ______.
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 ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
Evaluate: `int_0^π x/(1 + sinx)dx`.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
`int_-9^9 x^3/(4-x^2) dx` =______
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
