What is the integral from a to x of f(t) ?

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

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integral from a to x of f(t)

\int_{a}^{x} f (t) d t

\frac{f ( x ^{2} - a ^{2} )}{2}

Solution steps

\int_{a}^{x} f t d t

Take the constant out:

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

= f \cdot \int_{a}^{x} t d t

Apply the Power Rule

: [\frac{t ^{2}}{2}]_{a}^{x}

= f [\frac{t ^{2}}{2}]_{a}^{x}

Compute the boundaries:

\frac{x ^{2}}{2} - \frac{a ^{2}}{2}

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

Simplify

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

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