Advertisements
Advertisements
प्रश्न
Evaluate : `int_-1^1 log ((2 - x)/(2 + x))dx`.
Advertisements
उत्तर
Let f(x) = `log((2 - x)/(2 + x))`
We have, f(– x) = `log((2 + x)/(2 - x))`
= `-log((2 - x)/(2 + x))`
= – f(x)
So, f(x) is an odd function.
∴ `int_-1^1 log ((2 - x)/(2 + x))dx` = 0.
APPEARS IN
संबंधित प्रश्न
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) (2log sin x - log sin 2x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
`int_2^4 x/(x^2 + 1) "d"x` = ______
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`
The value of `int_-3^3 ("a"x^5 + "b"x^3 + "c"x + "k")"dx"`, where a, b, c, k are constants, depends only on ______.
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_0^1 x tan^-1x dx` = ______
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
Show that `int_0^(pi/2) (sin^2x)/(sinx + cosx) = 1/sqrt(2) log (sqrt(2) + 1)`
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
Evaluate: `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7) - x)dx`
The integral `int_0^2||x - 1| -x|dx` is equal to ______.
The value of the integral `int_0^sqrt(2)([sqrt(2 - x^2)] + 2x)dx` (where [.] denotes greatest integer function) is ______.
`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 ______.
`int_0^(π/2)((root(n)(secx))/(root(n)(secx + root(n)("cosec" x))))dx` is equal to ______.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
`int_0^(2a)f(x)/(f(x)+f(2a-x)) dx` = ______
Evaluate `int_0^3root3(x+4)/(root3(x+4)+root3(7-x)) dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Solve.
`int_0^1e^(x^2)x^3dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate:
`int_0^sqrt(2)[x^2]dx`
Which expression equals \[\int_{a}^{b} f(x)\,dx\] by \[P_3\] : The "King's Rule"?
