Evaluate the definite integral using the properties of even and odd functions.

S 2 (1/2 t^4+3)dt
-2

I just did.

http://www.jiskha.com/display.cgi?id=1447178366#1447178366.1447179649

(1/10) t^5 + 3 t

both are odd so the answer from -2 to +2 is twice answer from 0 to 2

2 [ 2^5/10 + 6 ]

2 ( 9.2) = 18.4

To evaluate the definite integral, we first need to understand the properties of even and odd functions.

An even function is symmetric about the y-axis, meaning that for every value of x, f(-x) = f(x). This property allows us to simplify the integral when the function is even.

An odd function, on the other hand, is symmetric about the origin, meaning that for every value of x, f(-x) = -f(x). This property also allows us to simplify the integral when the function is odd.

Now, let's evaluate the given definite integral using the properties of even and odd functions.

∫[from -2 to 2] (1/2 t^4 + 3) dt

Step 1: Separate the integral into two parts - one for the even function and one for the constant term.

∫[from -2 to 2] (1/2 t^4) dt + ∫[from -2 to 2] 3 dt

Step 2: For the first part, the function t^4 is an even function because it is a polynomial with an even power. We can take advantage of its symmetry and rewrite the integral.

∫[from -2 to 2] (1/2 t^4) dt = 2∫[from 0 to 2] (1/2 t^4) dt

Step 3: Simplify the first part of the integral:

2∫[from 0 to 2] (1/2 t^4) dt = ∫[from 0 to 2] t^4 dt

Step 4: Evaluate the integral of t^4:

∫[from 0 to 2] t^4 dt = (1/5) t^5 | [from 0 to 2]
= (1/5) (2^5 - 0^5)
= (1/5) (32 - 0)
= 32/5

Step 5: Simplify the second part of the integral, which is a constant:

∫[from -2 to 2] 3 dt = 3∫[from -2 to 2] dt

Step 6: Evaluate the integral of a constant:

3∫[from -2 to 2] dt = 3 ( t ) | [from -2 to 2]
= 3 ( 2 - (-2) )
= 3 ( 4 )
= 12

Step 7: Add the results of the two parts to find the overall result of the integral:

32/5 + 12 = 32/5 + 60/5 = 92/5

Therefore, the value of the definite integral ∫[from -2 to 2] (1/2 t^4 + 3) dt is 92/5.