Question

\( \int_1^1 (x^{19} + x^{21}) \, dx = 0 \)

Hint:

Whenever the limits of a definite integral are the same, the value of the integral is always zero, regardless of the function being integrated.

Answer 1

Step 1: Understanding the property of definite integrals.
When evaluating a definite integral where the limits of integration are the same (in this case, from 1 to 1), the integral always equals zero. This is because the area under the curve between identical limits is zero. Step 2: Verifying with the given function.
The function \( x^{19} + x^{21} \) is continuous, and we know that: \[ \int_1^1 f(x) \, dx = 0 \] for any continuous function \( f(x) \), as the lower and upper limits of the integration are the same. Conclusion:
Thus, the statement is True because the integral from 1 to 1 of any function is always zero.