Question

Is this true or false ??

Answer 1

Answer:  True

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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.

Answer 2

True
Good job and hope you have a good day