Question

\( \int_a^b f(x) \, dx \) is equal to:

• A. \( \int_a^b f(a - x) \, dx \)
• B. \( \int_a^b f(a + b - x) \, dx \)
• C. \( \int_a^b f(x - (a + b)) \, dx \)
• D. \( \int_a^b f((a - x) + (b - x)) \, dx \)

Hint:

In definite integrals, substitution with a linear transformation often results in limits being swapped, simplifying the integral.

Answer 1

The Correct Option is B

To assess the transformation, define \( u = a + b - x \).

The derivative is: \[ \frac{du}{dx} = -1 \quad \text{which implies} \quad dx = -du. \] The limits change as follows: when \( x = a \), \( u = b \); and when \( x = b \), \( u = a \). 

The integral transforms to: \[ \int_a^b f(x) \, dx = \int_b^a f(a + b - u) (-du). \] 

By inverting the integration limits, the negative sign is eliminated: \[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx. \]