Advertisements
Advertisements
प्रश्न
Prove that `int_0^af(x)dx=int_0^af(a-x) dx`
hence evaluate `int_0^(pi/2)sinx/(sinx+cosx) dx`
उत्तर
Let I = `int_0^af(x)dx`
Put x = a – t
∴ dx = – dt
When x = 0, t = a - 0 = a
When x = a, t = a - a = 0
`I=int_0^af(x)dx=int_a^0f(a-t)(-dt)`
`=-int_a^0f(a-t)dt` ......[`∵int_a^bf(x)dx=-int_b^af(x)dx`]
`=int_0^af(a-x)dx` ......[`∵int_a^bf(x)dx=-int_b^af(t)dx`]
`therefore int_0^af(x)dx=int_0^af(a-x)dx`
Let I=`int_0^(pi/2)sinx/(sinx+cosx)` ........(i)
`I=int_0^(pi/2)sin(pi/2-x)/(sin(pi/2-x)+cos(pi/2-x))` ......[`∵int_0^af(x)dx=-int_0^af(a-x)dx`]
`=int_0^(pi/2)cosx/(cosx+sinx)dx` ............(ii)
Adding (i) and (ii), we get
`2I=int_0^(pi/2)(sinx+cosx)/(sinx+cosx)dx`
`=int_0^(pi/2)1 dx`
`=[x]_0^(pi/2)`
`=pi/2-0`
`2I=pi/2`
`I=pi/4`
`therefore int_0^(pi/2)sinx/(sinx+cosx)dx=pi/4`
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) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Evaluate : \[\int(3x - 2) \sqrt{x^2 + x + 1}dx\] .
Evaluate = `int (tan x)/(sec x + tan x)` . dx
`int_1^2 1/(2x + 3) dx` = ______
`int_0^{pi/2} log(tanx)dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_0^pi sin^2x.cos^2x dx` = ______
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^pi x sin^2x dx` = ______
Evaluate `int_0^(pi/2) (tan^7x)/(cot^7x + tan^7x) "d"x`
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 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) = ______.
Evaluate the following:
`int_0^(pi/2) "dx"/(("a"^2 cos^2x + "b"^2 sin^2 x)^2` (Hint: Divide Numerator and Denominator by cos4x)
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
Evaluate: `int_0^π 1/(5 + 4 cos x)dx`
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
If `int_0^K dx/(2 + 18x^2) = π/24`, then the value of K is ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
If `int_0^(2π) cos^2 x dx = k int_0^(π/2) cos^2 x dx`, then the value of k is ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
`int_1^2 x logx dx`= ______
Evaluate the following definite integral:
`int_-2^3 1/(x + 5) dx`
Evaluate:
`int_0^1 |2x + 1|dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) 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`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
