finding the exact value of the integral?

Question

hi. i am asking this question one more time as every time beofre that i have asked it i get a repsonse that the answer is 0. i have been told by a maths person it is not 0

i would like to find out how to solve this problem

i would like to evaluate the integral -5 to 5 | sqrt 3x^7+x^1/7| dx

the bar represents absolute value

the maths person said it was not 0 because

Yes the integrand is the absolute value of a function. Let's call that
function inside the absolute value signs, f(x).
Now f(x) itself is an odd function, which is to say that it satisfies:
f(-x) = -f(x)
However, f(x) is positive for x > 0.
So, for x > 0 (so that -x < 0) we have:

|f(-x)| = |-f(x)| = |f(x)|

which is to say that the absolute value of an *odd* function is *even*,
Now, when you integrate an *odd* function between symmetric limits, i.e.between -T and T for some real number T, the integral from -T to 0 cancelsout the integralfrom 0 to T ... and so the result is 0.
ths for any help

When you integrate an *even* function between symmetric limits, the
integral from -T to 0 equals the integral from 0 to T ... and so the
result is *twice* the integral from 0 to T.

Now since f(x) is positive for x > 0, you have |f(x)| = f(x) for x > 0,
which means your integral becomes:

T T
S |f(x)| dx = 2 S |f(x)| dx
-T 0
T
= 2 S f(x) dx
0
That's what you were supposed to `see' when doing this question.

Anyway, the answer is assuredly not zero.

the bit above was the rest of what he said.... to reiterate the answer is not 0.

Answer 1

You don't make clear what part of the integrand the square root applies to. I'll assume it is just the 3, i.e. (√3) x^7 rather than √(3x^7), because in the latter case the square root is undefined when x is negative.

Note that both terms in the integrand are negative precisely when x is negative, so the whole integrand is negative on [-5, 0) and positive on (0, 5]. As mentioned, the integrand is even, so we know

∫(-5 to 5) |(√3)x^7+x^(1/7)| dx
= 2 ∫(0 to 5) |(√3)x^7+x^(1/7)| dx
= 2 ∫(0 to 5) (√3)x^7+x^(1/7) dx (since the integrand is positive here)
= 2 [(√3)x^8 / 8 + x^(8/7) / (8/7)][0 to 5]
= 2 [(√3)5^8 / 8 + 5^(8/7) (7/8) - 0]
= 5^8 √3 / 4 + 7 . 5^(8/7) / 4.
≈ 169156.6.

Answer 2

It seems like unless your absolute value bars are mislocated, you have a domain problem. The function is discontinuous within your bounds of integration.

Answer 3

tan^2x = sin^2x/cos^2x = [a million - cos^2x]/cos^2x = a million/cos^2x - a million = sec^2x - a million critical = tanx - x between limits [tanpi/3 - pi/3] - [tanpi/6 - pi/6] = [tan60 - tan30 - pi/6] = 0.631102 {if pi/6 on incredible the respond is - 0.631102}