determining the value of an integarl given a certain interval?

Question

if int -1 to 5 f(x)dx = 3.7 and int -1 to 3 f(x) dx = 3.2
what is the value of the int 3 to 5 (3f(x)+2x)dx?

Answer 1

In order to calculate

Integral (3 to 5, 3f(x) + 3x)dx

You have to note that the integral of the sum is equal to the sum of integrals. In other words, we can split that integral up, into

Integral (3 to 5, 3f(x))dx + Integral (3 to 5, 3x)dx

Not only can we split integrals up, but we can also pull out constants. In the first integral, there's a constant 3, and the second, a constant 2.

3 * Integral (3 to 5, f(x)dx) + 2 * Integral (3 to 5, x dx)

The integral from -1 to 5 of f(x) minus the integral of -1 to 3 of f(x) will give the remaining interval, the integral of 3 to 5 of f(x). So we just simply subtract:

3.7 - 3.2 = 0.5

Therefore,
Integral (3 to 5, f(x))dx = 0.5

Now that we have this value, we just plug in 0.5 for every occurrance of Integral (3 to 5, f(x))dx.

So for here:

3 * Integral (3 to 5, f(x)dx) + 2 * Integral (3 to 5, x dx)

We substitute appropriately.

3 [0.5] + 2 * Integral (3 to 5, x) dx

The Integral of x is easy to solve; it's just (x^2)/2. So we just evaluate (x^2)/2 from 3 to 5.

1.5 + 2 * [ (x^2)/2, evaluated from 3 to 5 ]
1.5 + 2 * [25/2 - 9/2]
1.5 + 2 * [16/2]
1.5 + 2 [8]
1.5 + 16 = 17.5

Answer 2

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Answer 3

17.5