What is the integral from a to b of f(a+b-x) ?

The integral from a to b of f(a+b-x) is f(-a^2-(-a^2+b^2)/2+b^2)

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integral from a to b of f(a+b-x)

\int_{a}^{b} f (a + b - x) d x

f (- a^{2} - \frac{- a ^{2} + b ^{2}}{2} + b^{2})

Solution steps

\int_{a}^{b} f (a + b - x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= f \cdot \int_{a}^{b} a + b - x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= f (\int_{a}^{b} a d x + \int_{a}^{b} b d x - \int_{a}^{b} x d x)

\int_{a}^{b} a d x = ab - a^{2}

\int_{a}^{b} b d x = b^{2} - ba

\int_{a}^{b} x d x = \frac{b ^{2} - a ^{2}}{2}

= f (ab - a^{2} + b^{2} - ba - \frac{b ^{2} - a ^{2}}{2})

Simplify

= f (- a^{2} - \frac{- a ^{2} + b ^{2}}{2} + b^{2})

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What is the integral from a to b of f(a+b-x) ?
The integral from a to b of f(a+b-x) is f(-a^2-(-a^2+b^2)/2+b^2)