find value of def integral with a=-2 and b=2sqrt(3)
for integral i get 1/15 *((4+x^2)^(3/2)) (-8+3x^2)
for value i get [1536- 64sqrt(2)]/15 but its' wrong. help please
To find the value of a definite integral, you need to substitute the upper and lower limits of integration into the antiderivative expression and then calculate the difference between the two results.
The given antiderivative is (1/15) * ((4 + x^2)^(3/2) * (-8 + 3x^2)).
To calculate the definite integral with the limits a = -2 and b = 2√3, you need to substitute these values into the antiderivative expression and subtract the result at a from the result at b.
Substituting a = -2:
(1/15) * ((4 + (-2)^2)^(3/2) * (-8 + 3(-2)^2))
= (1/15) * ((4 + 4)^(3/2) * (-8 + 12))
= (1/15) * (8^(3/2) * 4)
= (1/15) * (64 * 4)
= 256/15
Substituting b = 2√3:
(1/15) * ((4 + (2√3)^2)^(3/2) * (-8 + 3(2√3)^2))
= (1/15) * ((4 + 12)^(3/2) * (-8 + 3(12)))
= (1/15) * (16^(3/2) * (-8 + 36))
= (1/15) * (64 * 28)
= 1792/15
Finally, subtracting the result at a from the result at b:
(1792/15) - (256/15) = 1536/15
Therefore, the correct value of the definite integral is 1536/15, which simplifies to 102.4.