Question

Evaluate the integral: \[ \int_{\alpha}^{\beta} \varphi(x) \, dx + \int_{\beta}^{\alpha} \varphi(x) \, dx \]

• A. 2
• B. 1
• C. 0
• D. \( 2 \int_{\alpha}^{\beta} \varphi(x) \, dx \)

Hint:

When adding integrals with reversed limits, the sum will always be zero.

Answer 1

The Correct Option is C

We are given the sum of two integrals, \( \int_{\alpha}^{\beta} \varphi(x) \, dx + \int_{\beta}^{\alpha} \varphi(x) \, dx \). By the property of definite integrals, we know that: \[ \int_{\alpha}^{\beta} \varphi(x) \, dx = -\int_{\beta}^{\alpha} \varphi(x) \, dx. \] Thus, adding these two integrals together gives: \[ \int_{\alpha}^{\beta} \varphi(x) \, dx + \int_{\beta}^{\alpha} \varphi(x) \, dx = 0. \] Hence, the correct answer is option (C) \( 0 \).