Advertisements
Advertisements
Question
Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`
Advertisements
Solution 1
I = `int_1^4 (|x -1|+|x - 2|+|x - 4|)dx`
Let f (x) = |x - 1| + |x - 2| + |x - 4|
We have three critical points x = 1, 2, 4
(i) when x <1
(ii) when 1≤ x < 2
(iii) when 2 ≤ x < 4
(iv) when x ≥ 4
Solution 2
RELATED QUESTIONS
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) sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx`
By using the properties of the definite integral, evaluate the integral:
`int_(-5)^5 | x + 2| 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_((-pi)/2)^(pi/2) sin^2 x dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^a sqrtx/(sqrtx + sqrt(a-x)) dx`
Evaluate`int (1)/(x(3+log x))dx`
Evaluate : `int 1/sqrt("x"^2 - 4"x" + 2) "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: `int_0^pi ("x"sin "x")/(1+ 3cos^2 "x") d"x"`.
`int_1^2 1/(2x + 3) dx` = ______
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_0^{1/sqrt2} (sin^-1x)/(1 - x^2)^{3/2} dx` = ______
`int_0^pi x*sin x*cos^4x "d"x` = ______.
`int_0^(pi/2) cos x "e"^(sinx) "d"x` is equal to ______.
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`
`int_0^1|3x - 1|dx` equals ______.
If f(x) = `{{:(x^2",", "where" 0 ≤ x < 1),(sqrt(x)",", "when" 1 ≤ x < 2):}`, then `int_0^2f(x)dx` equals ______.
If `int_0^(π/2) log cos x dx = π/2 log(1/2)`, then `int_0^(π/2) log sec dx` = ______.
`int_((-π)/2)^(π/2) log((2 - sinx)/(2 + sinx))` is equal to ______.
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`.
Evaluate: `int_0^(π/4) log(1 + tanx)dx`.
Evaluate the following definite integral:
`int_4^9 1/sqrt"x" "dx"`
If `int_0^1(3x^2 + 2x+a)dx = 0,` then a = ______
Solve the following.
`int_0^1e^(x^2)x^3 dx`
Evaluate the following integral:
`int_0^1 x(1 - x)^5 dx`
Evaluate the following integral:
`int_0^1x(1-x)^5dx`
Evaluate the following definite intergral:
`int_1^3logx dx`
`int_0^(pi/4) (cos^2 x)/(cos^2 x + 4 sin^2 x) dx` =
