Definite Integral

**softstyll** · December 11th 2011, 02:30 PM

integral from -1 to 1 of (x^99)*cosh(x)dx

apparently, the answer is obtained very easily with minimal work...i don't see it....

**Prove It** · December 11th 2011, 02:41 PM

Originally Posted by softstyll

integral from -1 to 1 of (x^99)*cosh(x)dx

apparently, the answer is obtained very easily with minimal work...i don't see it....

$\displaystyle f(x) &= x^{99}\cosh{x} \\ f(-x) &= (-x)^{99}\cosh{(-x)} \\ f(-x) &= -x^{99}\cosh{x} \\ f(-x) &= -f(x)$

This is an odd function. What do you know about definite integrals of odd functions?

**softstyll** · December 11th 2011, 02:51 PM

Originally Posted by Prove It

$\displaystyle f(x) &= x^{99}\cosh{x} \\ f(-x) &= (-x)^{99}\cosh{(-x)} \\ f(-x) &= -x^{99}\cosh{x} \\ f(-x) &= -f(x)$

This is an odd function. What do you know about definite integrals of odd functions?

Thank you. I understand that the integral of an odd function from -a to a is a constant. But I don't understand how u can just say that f(x) is the integrand.

**Prove It** · December 11th 2011, 02:56 PM

Originally Posted by softstyll

Thank you. I understand that the integral of an odd function from -a to a is a constant. But I don't understand how u can just say that f(x) is the integrand.

Yes it is a constant, in fact, it's 0.

Maybe the reason I know that f(x) is the integrand is because you told me what the integrand was... ><

**Plato** · December 11th 2011, 03:00 PM

Originally Posted by softstyll

Thank you. I understand that the integral of an odd function from -a to a is a constant. But I don't understand how u can just say that f(x) is the integrand.

Do you understand $\int_{ - 99}^{99} {(x^3 + \sin ^{999} (x))dx} = 0$ and WHY?

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