Advertisements
Advertisements
प्रश्न
Evaluate: `int_(-1)^3 |x^3 - x|dx`
Advertisements
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
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
Evaluate : `int _0^(pi/2) "sin"^ 2 "x" "dx"`
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
`int_0^2 e^x dx` = ______.
`int_0^(pi"/"4)` log(1 + tanθ) dθ = ______
`int_0^{pi/2}((3sqrtsecx)/(3sqrtsecx + 3sqrt(cosecx)))dx` = ______
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
`int_0^1 (1 - x)^5`dx = ______.
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
`int_(pi/4)^(pi/2) sqrt(1-sin 2x) dx =` ______.
`int_0^{pi/2} (cos2x)/(cosx + sinx)dx` = ______
Evaluate `int_(-1)^2 "f"(x) "d"x`, where f(x) = |x + 1| + |x| + |x – 1|
`int_(-2)^2 |x cos pix| "d"x` is equal to ______.
`int_(-"a")^"a" "f"(x) "d"x` = 0 if f is an ______ function.
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) sqrt(1 - sin2x) "d"x` is equal to ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
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 ______.
`int_0^(π/2)((root(n)(secx))/(root(n)(secx + root(n)("cosec" x))))dx` is equal to ______.
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Evaluate the following definite integral:
`int_-2^3(1)/(x + 5) dx`
