Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| dx`
Advertisements
उत्तर
Let `I = int_-5^5 abs (x + 2) dx`
Define,
`abs (x + 2) = {(-(x + 2), if x + 2 < 0, or x< - 2),(x + 2, if x +2 >= 0, or x >=-2):}`
∵ `I = - int_-5^-2 (x + 2) dx + int_-2^5 (x + 2) dx`
`= -[(x + 2)^2/2]_-5^-2 + [(x + 2)^2/2]_-2^5`
`= [((-2 + 2)^2/2 - (-5 + 2)^2/2)] + [(5 + 2)^2/2 - (-2 + 2)^2/2]`
`= -1/2 [-9] + 1/2 [49 - 0]`
`= 9/2 + 49/2`
`= 58/2`
= 29
APPEARS IN
संबंधित प्रश्न
Evaluate : `intlogx/(1+logx)^2dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) cos^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/4) log (1+ tan x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (2log sin x - log sin 2x)dx`
Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
Prove that `int _a^b f(x) dx = int_a^b f (a + b -x ) dx` and hence evaluate `int_(pi/6)^(pi/3) (dx)/(1 + sqrt(tan x))` .
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Find `dy/dx, if y = cos^-1 ( sin 5x)`
Evaluate : ∫ log (1 + x2) dx
Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
`int_"a"^"b" "f"(x) "d"x` = ______
`int_0^1 "e"^(2x) "d"x` = ______
`int_1^2 1/(2x + 3) dx` = ______
`int_0^{pi/2} xsinx dx` = ______
If `int_0^"a" sqrt("a - x"/x) "dx" = "K"/2`, then K = ______.
`int_-2^1 dx/(x^2 + 4x + 13)` = ______
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
`int_-1^1x^2/(1+x^2) dx=` ______.
`int_0^(pi/2) 1/(1 + cos^3x) "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:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
Evaluate: `int_(-1)^3 |x^3 - x|dx`
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
Let `int ((x^6 - 4)dx)/((x^6 + 2)^(1/4).x^4) = (ℓ(x^6 + 2)^m)/x^n + C`, then `n/(ℓm)` is equal to ______.
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`
Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.
Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.
The value of `int_0^(π/4) (sin 2x)dx` is ______.
Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Evaluate the following integral:
`int_-9^9 x^3/(4-x^2)dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
