Question

Is this true or false ??

Answer 1

True
Good job and hope you have a good day

Answer 2

Answer: True

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Step-by-step explanation:

We'll use these two properties of integrals
`\displaystyle \text{If f(x) is an even function, then } \int_(-a)^(a)f(x)dx = 2\int_(0)^(a)f(x)dx`

`\displaystyle \text{If f(x) is an odd function, then } \int_(-a)^(a)f(x)dx = 0`

These properties are valid simply because of the function's symmetry. For even functions, we have vertical axis symmetry about x = 0; while odd functions have symmetry about the origin.

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Here's how the steps could look

`\displaystyle \int_(-7)^(7)(ax^8+bx+c)dx=\int_(-7)^(7)((ax^8+c)+bx)dx\\\\\\\displaystyle \int_(-7)^(7)(ax^8+bx+c)dx=\int_(-7)^(7)(ax^8+c)dx+\int_(-7)^(7)(bx)dx\\\\\\\displaystyle \int_(-7)^(7)(ax^8+bx+c)dx=\left(2\int_(0)^(7)(ax^8+c)dx\right)+(0)\\\\\\\displaystyle \int_(-7)^(7)(ax^8+bx+c)dx=2\int_(0)^(7)(ax^8+c)dx\\\\\\`

Therefore, the given statement is true. The values of a,b,c don't matter. You could replace those '7's with any real number you want and still end up with a true statement.

We can see that ax^8+c is always even, while bx is always odd.

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Side note:

For the second step, I used the idea that
`\int(f(x)+g(x))dx=\int f(x)dx+\int g(x)dx\\\\`

which allows us to break up a sum into smaller integrals.