Advertisements
Advertisements
प्रश्न
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
Advertisements
उत्तर
`int_0^4 abs (x - 1) dx`
Define,
`abs(x - 1) = {(-(x-1), if x-1<0, or x < 1),(x-1, if x - 1>=0, or x>=1):}`
`int_0^1 abs (x - 1) dx + int_1^4 abs(x - 1) dx`
`int_0^1 - (x - 1) "dx" + int_1^4 (x - 1) dx`
`= - [x^2/2 - x]_0^1 + [x^2/2 - x]_1^4`
`= [(1/2 - 1) - 0] + (16/2 - 4) - (1/2 - 1)`
`= 1/2 + 4 + 1/2`
`= (1 + 8 + 1)/2`
= 5
APPEARS IN
संबंधित प्रश्न
Evaluate : `int e^x[(sqrt(1-x^2)sin^-1x+1)/(sqrt(1-x^2))]dx`
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/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) 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_2^8 |x - 5| dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi (x dx)/(1+ sin x)`
By using the properties of the definite integral, evaluate the integral:
`int_0^(2x) cos^5 xdx`
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.
If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that
Evaluate : `int "e"^(3"x")/("e"^(3"x") + 1)` dx
Evaluate : ∫ log (1 + x2) dx
`int_0^{pi/4} (sin2x)/(sin^4x + cos^4x)dx` = ____________
`int_0^{pi/2} cos^2x dx` = ______
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
`int_{pi/6}^{pi/3} sin^2x dx` = ______
If `int_0^"k" "dx"/(2 + 32x^2) = pi/32,` then the value of k is ______.
The value of `int_2^7 (sqrtx)/(sqrt(9 - x) + sqrtx)dx` is ______
Find `int_0^(pi/4) sqrt(1 + sin 2x) "d"x`
`int_((-pi)/4)^(pi/4) "dx"/(1 + cos2x)` is equal to ______.
`int_0^(pi/2) sqrt(1 - sin2x) "d"x` is equal to ______.
If `int (log "x")^2/"x" "dx" = (log "x")^"k"/"k" + "c"`, then the value of k is:
Evaluate: `int_(-1)^3 |x^3 - x|dx`
`int_0^1 1/(2x + 5) dx` = ______.
`int_4^9 1/sqrt(x)dx` = ______.
`int_0^π(xsinx)/(1 + cos^2x)dx` equals ______.
The value of the integral `int_0^1 x cot^-1(1 - x^2 + x^4)dx` is ______.
`int_(π/3)^(π/2) x sin(π[x] - x)dx` is equal to ______.
Evaluate `int_-1^1 |x^4 - x|dx`.
Evaluate the following integral:
`int_0^1 x(1 - 5)^5`dx
Evaluate the following integral:
`int_0^1x (1 - x)^5 dx`
Evaluate the following integral:
`int_-9^9 x^3/(4 - x^2) dx`
Solve the following.
`int_0^1 e^(x^2) x^3dx`
Evaluate the following integral:
`int_0^1 x(1-x)^5 dx`
Evaluate:
`int_0^6 |x + 3|dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
The value of \[\int_{-1}^{1}\left(\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\right)\mathrm{d}x\] is
What is the value of \[\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\sin^{2}x\,dx\]?
