Advertisements
Advertisements
प्रश्न
Evaluate: `int_(-1)^3 |x^3 - x|dx`
उत्तर
Let I = `int_(-1)^2|x^3 - x|dx`
= `int_(-1)^2|x(x^2 - 1)|dx`
= `int_(-1)^2|x(x - 1)(x + 1)|dx`
Here, x3 – x = 0, when x = 0, 1, –1
| Value of x | Value of (x3 – x) |
| –1 < x < 0 | +ve |
| 0 < x < 1 | –ve |
| 1 < x < 2 | +ve |
∴ |x3 – x| = `{{:(x^3 - x, if -1 < x < 0 and 1 < x < 2),(-x^3 + x, if 0 < x < 1):}`
I = `int_(-1)^0(x^3 - x)dx + int_1^1(-x^3 + x)dx + int_1^2(x^3 - x)dx`
= `[x^4/4 - x^2/2]_-1^0 + [(-x^4)/4 + x^2/2]_0^1 + [x^4/4 - x^2/2]_1^2`
= `1/4 + 1/4 + 2 + 1/4`
= `2 + 3/4`
= `11/4`
APPEARS IN
संबंधित प्रश्न
Evaluate :`int_0^pi(xsinx)/(1+sinx)dx`
By using the properties of the definite integral, evaluate the integral:
`int_((-pi)/2)^(pi/2) sin^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_(pi/2)^(pi/2) sin^7 x dx`
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Evaluate : `int 1/("x" [("log x")^2 + 4]) "dx"`
`int_0^1 "e"^(2x) "d"x` = ______
`int_(-7)^7 x^3/(x^2 + 7) "d"x` = ______
`int (cos x + x sin x)/(x(x + cos x))`dx = ?
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
`int_0^(pi/2) 1/(1 + cosx) "d"x` = ______.
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
`int_0^(pi/2) (sin^"n" x"d"x)/(sin^"n" x + cos^"n" 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)
Evaluate the following:
`int_(-pi/4)^(pi/4) log|sinx + cosx|"d"x`
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
If `int_0^"a" 1/(1 + 4x^2) "d"x = pi/8`, then a = ______.
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 ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
Evaluate `int_-1^1 |x^4 - x|dx`.
Evaluate: `int_0^π x/(1 + sinx)dx`.
For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x dx` is ______.
`int_1^2 x logx dx`= ______
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Evaluate:
`int_0^6 |x + 3|dx`
Evaluate the following definite intergral:
`int_1^2 (3x)/(9x^2 - 1) dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following integral:
`int_0^1x(1 - x)^5dx`
