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 evaluate the transformation, let $u = a + b - x$. 

Then: $$\frac{du}{dx} = -1 \quad \text{or} \quad dx = -du.$$ When $x = a$, $u = b$; and when $x = b$, $u = a$. 

The integral becomes: $$\int_a^b f(x) \, dx = \int_b^a f(a + b - u) (-du).$$ 

Reversing the limits of integration, the negative sign is removed: $$\int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx.$$