Integral Calculator With L And E

Integral of e^x:

\[ \int e^x \, dx = e^x + C \]

Result:

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1. What is the Integral of e^x?

The integral of e^x is unique because it's one of the few functions whose derivative and integral are the same. The general form is:

\[ \int e^x \, dx = e^x + C \]

For exponential functions with coefficients in the exponent:

\[ \int e^{ax} \, dx = \frac{1}{a}e^{ax} + C \]

2. How Does the Calculator Work?

The calculator handles both indefinite and definite integrals of exponential functions:

For indefinite: \[ \int e^{ax} \, dx = \frac{1}{a}e^{ax} + C \]

For definite: \[ \int_{lower}^{upper} e^{ax} \, dx = \frac{e^{a\cdot upper} - e^{a\cdot lower}}{a} \]

3. Importance of Exponential Integrals

Applications: Exponential integrals appear in growth/decay problems, probability, physics (radioactive decay), finance (continuous compounding), and differential equations.

4. Using the Calculator

Instructions: Enter the coefficient for the exponent (default is 1 for e^x). For definite integrals, provide both lower and upper limits.

5. Frequently Asked Questions (FAQ)

Q1: Why is the integral of e^x itself?
A: This is a unique property of the exponential function with base e, as the slope of e^x at any point equals its value at that point.

Q2: What if the exponent coefficient is zero?
A: e^0 = 1, so the integral becomes ∫1 dx = x + C. The calculator handles this case separately.

Q3: Can this calculator handle e^(x^2)?
A: No, e^(x^2) doesn't have an elementary antiderivative. Its integral requires special functions or numerical methods.

Q4: What about integrals like ∫x e^x dx?
A: These require integration by parts. This calculator currently handles only simple exponential integrals.

Q5: How precise are the calculations?
A: Results are accurate to 4 decimal places for definite integrals.