Primitive of Exponential Function/Examples/1 - x
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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 $\text {XIV}$: $4$.