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The given input appears to be an incomplete expression for a definite integral. To make it a complete and meaningful question, we need to specify the function being integrated. Here is the corrected and completed version:

Evaluate the definite integral:

[tex]\[ \int_{x+9}^{2x+4} f(t) \, dt \][/tex]

where [tex]\( f(t) \)[/tex] is the function being integrated.

If additional context or the function [tex]\( f(t) \)[/tex] was intended to be provided, please include it for a more accurate representation.



Answer :

GinnyAnswer
Sure! Let's evaluate the integral of a given function with respect to [tex]\( x \)[/tex] over the interval [tex]\([x+9, 2x+4]\)[/tex].

The integral to evaluate is:

[tex]\[ \int_{x+9}^{2x+4} 1 \, dx \][/tex]

Here, the integrand is the constant function [tex]\( 1 \)[/tex]. When integrating a constant, the result is the constant multiplied by the length of the integration interval. Therefore, we need to determine the length of the interval from [tex]\( x+9 \)[/tex] to [tex]\( 2x+4 \)[/tex].

The length of the interval can be found by subtracting the lower bound from the upper bound:

[tex]\[ \text{Length of interval} = (2x + 4) - (x + 9) \][/tex]

Simplify the expression inside the parentheses:

[tex]\[ (2x + 4) - (x + 9) = 2x + 4 - x - 9 \][/tex]
[tex]\[ = x - 5 \][/tex]

Now, since the integrand is [tex]\( 1 \)[/tex], the integral of [tex]\( 1 \)[/tex] over the interval [tex]\([x+9, 2x+4]\)[/tex] is simply the length of the interval:

[tex]\[ \int_{x+9}^{2x+4} 1 \, dx = x - 5 \][/tex]

So, the value of the given integral is:

[tex]\[ x - 5 \][/tex]

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