Step by step solution to "Suppose ∫−22f(x)dx=4,∫25f(x)dx=3,∫−25g(x)dx=2. Is the following statement true? ∫−25(f(x)+g(x))=9"

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sjeikdom0

Answered question

2021-05-31

Suppose $\int_{- 2}^{2} f (x) d x = 4, \int_{2}^{5} f (x) d x = 3, \int_{- 2}^{5} g (x) d x = 2$ .
Is the following statement true?
$\int_{- 2}^{5} (f (x) + g (x)) = 9$

Answer & Explanation

Aniqa O'Neill

Skilled2021-06-01Added 100 answers

Step 1
Given:
$\int_{- 2}^{2} f (x) d x = 4$
$\int_{2}^{5} f (x) d x = 3$
$\int_{- 2}^{5} g (x) d x = 2$
Step 2
Using properties of Definite Integrals:
$\int_{a}^{b} [f (x) \pm g (x)] d x = \int_{a}^{b} f (x) d x \pm \int_{a}^{b} g (x) d x$
$\int_{a}^{c} f (x) d x + \int_{c}^{b} f (x) d x = \int_{a}^{b} f (x) d x [c \in (a, b)]$
$\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x$
Step 3
Now,
$\int_{- 2}^{5} (f (x) + g (x)) d x = \int_{- 2}^{5} f (x) d x + \int_{- 2}^{5} g (x) d x$
$= \int_{- 2}^{2} f (x) d x + \int_{2}^{5} f (x) d x + \int_{- 2}^{5} g (x) d x$
$= 4 + 3 + 2$
$= 9$
True Statement

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