Primitive of Exponential Function/Examples/1 - x

Example of Use of Primitive of Exponential Function

$\ds \int e^{1 - x} \rd x = -e^{1 - x} + C$

Proof

From Primitive of Function of $a x + b$:

$\ds \int \map F {a x + b} \rd x = \frac 1 a \int \map F u \rd u$

where $u = a x + b$.

Hence:

\(\ds \ds \int e^{1 - x} \rd x\)

\(=\)

\(\ds \dfrac 1 {-1} \ds \int e^{1 - x} \map {\rd} {1 - x}\)

Primitive of Function of $a x + b$, setting $a \gets -1$, $b \gets 1$

\(\ds \)

\(=\)

\(\ds -e^{1 - x} + C\)

Primitive of Exponential Function

$\blacksquare$

Proof

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $4$.