Q. \[\int e^{x}\,dx\]
Answer
To integrate \(e^x\), use the fact that the derivative of \(e^x\) is \(e^x\). So the antiderivative is the same function.
\[
\int e^x\,dx = e^x + C
\]
Detailed Explanation
We want to compute the indefinite integral
\[
\int e^x \, dx
\]
Step 1: Recall a key derivative fact.
Differentiate \(e^x\):
\[
\frac{d}{dx}\left(e^x\right) = e^x
\]
This tells us that \(e^x\) is the derivative of \(e^x\).
Step 2: Use the inverse relationship between differentiation and integration.
If \(\frac{d}{dx}(e^x) = e^x\), then integrating \(e^x\) must return \(e^x\) (up to a constant).
Step 3: Add the constant of integration.
Because indefinite integrals always include a constant \(C\), the result is:
\[
\int e^x \, dx = e^x + C
\]
Final Answer:
\[
\int e^x \, dx = e^x + C
\]
See full solution
Calculus FAQ
What is the indefinite integral of \(e^x\) ?
\(\int e^x\,dx = e^x + C\).
What is the definite integral \(\int_0^1 e^x\,dx\) ?
\(\int_0^1 e^x\,dx = \left[e^x\right]_0^1 = e - 1\).
How to integrate \(e^{x+3}\) ?
\(\int e^{x+3}\,dx = e^{x+3} + C\). Because \(\frac{d}{dx}(x+3)=1\).
How to integrate \(e^{2x}\) ?
\(\int e^{2x}\,dx = \frac{1}{2}e^{2x} + C\), since \(\frac{d}{dx}(2x)=2\).
How to integrate \(e^{kx}\) for constant \(k\neq 0\) ?
\(\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C\).
What is \(\int e^x\,d x\) using substitution?
Let \(u=x\). Then \(\int e^x\,dx = \int e^u\,du = e^u + C = e^x + C\).
How to check the integral answer by differentiation?
Differentiate: \(\frac{d}{dx}\left(e^x + C\right)=e^x\), matching the integrand.
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