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Integration of Exponential Functions

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Mastering the integration of exponential functions is crucial in calculus. This includes understanding the antiderivative of natural exponential functions, such as e^x, and those with different bases, like a^x. Techniques like Integration by Substitution and Integration by Parts are vital for handling more complex functions, while avoiding common errors ensures accuracy. Definite and improper integrals are also discussed, highlighting the use of the Fundamental Theorem of Calculus and limits.

Fundamentals of Integrating Exponential Functions

Integration of exponential functions is a key operation in calculus, often more intricate than their differentiation. The natural exponential function, \( e^x \), is notable for its simple antiderivative, which is the function itself, \( \int e^x \, dx = e^x + C \). For exponential functions with a base other than \( e \), such as \( a^x \), the antiderivative includes a division by the natural logarithm of the base: \( \int a^x \, dx = \frac{1}{\ln(a)} a^x + C \), where \( a \) is a positive constant different from 1, and \( C \) is the constant of integration. It is essential to include \( +C \) to represent the general solution to the indefinite integral.
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Exponential Functions with Variable Exponents: Differentiation and Integration

Exponential functions with variable exponents, such as \( e^{2x^2} \), require the Chain Rule for differentiation. For example, if \( u = 2x^2 \), then \( \frac{df}{dx} = e^u \frac{du}{dx} \), with \( \frac{du}{dx} = 4x \), leading to \( \frac{df}{dx} = 4x e^{2x^2} \). Conversely, integrating such functions typically involves the method of Integration by Substitution. To integrate \( \int e^{3x} \, dx \), set \( u = 3x \) and \( du = 3 \, dx \), which implies \( dx = \frac{1}{3} \, du \). Substituting these into the integral yields \( \int e^{3x} \, dx = \frac{1}{3} \int e^u \, du = \frac{1}{3} e^u + C \), and reverting to \( x \), we obtain \( \frac{1}{3} e^{3x} + C \).

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00

Integration by Parts formula

Integrate u dv = uv - Integrate v du

01

Choosing u and dv in Integration by Parts

u = x^2, dv = e^x dx for Integrate x^2 e^x dx

02

Result of Integration by Parts for Integrate x^2 e^x dx

uv - Integrate v du yields x^2 e^x - 2 Integrate x e^x dx

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