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Suppose [tex]\int_1^3 f(x) \, dx=5[/tex], [tex]\int_3^8 f(x) \, dx=-4[/tex], and [tex]\int_3^8 g(x) \, dx=6[/tex]. Evaluate the integrals in parts a-d.

a. [tex]\int_1^3 6 f(x) \, dx = \square[/tex] (Simplify your answer.)

Sagot :

GinnyAnswer
To evaluate the integral [tex]\(\int_1^3 6 f(x) \, dx\)[/tex], we can use the property of integrals that allows us to factor out a constant. This property states that for any constant [tex]\(c\)[/tex],

[tex]\[ \int_a^b c \cdot f(x) \, dx = c \cdot \int_a^b f(x) \, dx. \][/tex]

Given the integral [tex]\(\int_1^3 f(x) \, dx = 5\)[/tex], we will apply this property with [tex]\(c = 6\)[/tex]:

[tex]\[ \int_1^3 6 f(x) \, dx = 6 \cdot \int_1^3 f(x) \, dx. \][/tex]

Substituting the given value of [tex]\(\int_1^3 f(x) \, dx\)[/tex]:

[tex]\[ \int_1^3 6 f(x) \, dx = 6 \cdot 5. \][/tex]

Thus,

[tex]\[ \int_1^3 6 f(x) \, dx = 30. \][/tex]

Therefore, the answer is [tex]\(30\)[/tex].
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