Advertisements
Advertisements
Question
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) cos^2 x dx`
Advertisements
Solution
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
RELATED QUESTIONS
Prove that: `int_0^(2a)f(x)dx=int_0^af(x)dx+int_0^af(2a-x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (cos^5 xdx)/(sin^5 x + cos^5 x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) 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`
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate`int (1)/(x(3+log x))dx`
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
Evaluate : ∫ log (1 + x2) dx
Evaluate `int_1^2 (sqrt(x))/(sqrt(3 - x) + sqrt(x)) "d"x`
Evaluate `int_1^3 x^2*log x "d"x`
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^{pi/2} xsinx dx` = ______
`int_0^1 (1 - x)^5`dx = ______.
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_0^1 log(1/x - 1) "dx"` = ______.
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
`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`
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.
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_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_a^b f(x)dx = int_a^b f(x - a - b)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) = `(2 - xcosx)/(2 + xcosx)` and g(x) = logex, (x > 0) then the value of the integral `int_((-π)/4)^(π/4) "g"("f"(x))"d"x` is ______.
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`
With the usual notation `int_1^2 ([x^2] - [x]^2)dx` is equal to ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
`int_-1^1 (17x^5 - x^4 + 29x^3 - 31x + 1)/(x^2 + 1) dx` is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
Evaluate the following limit :
`lim_("x"->3)[sqrt("x"+6)/"x"]`
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate the following integral:
`int_0^1 x (1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9x^3/(4-x^2)dx`
