Question

Find the following. $$\int 3^{2 x} d x$$

   Find the following.
$$\int 3^{2 x} d x$$

Calculus with Applications

Calculus with Applications

Margaret L. Lial •… 11th Edition

Chapter 7, Problem 42

Instant Answer

Solved by Expert Bobby Barnes

04/11/2019

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In this case, the base is 3 and the exponent is 2x. Show more…

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Find the following. $$\int 3^{2 x} d x$$

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Key Concepts

Exponential Functions

Exponential functions take the form a^(x), where a is a constant. They are fundamental in mathematics due to their distinct growth properties and appear frequently in calculus problems involving differentiation and integration.

Integration of Exponential Functions

The integral of an exponential function with a constant base, such as a^(x), is typically found using the formula ? a^(x) dx = a^(x)/(ln a) + C. This formula is derived from the derivative of a^(x), which is a^(x) multiplied by the natural logarithm of the base a.

Substitution Method (u-substitution)

When the exponent of an exponential function is a linear function of x, such as in a^(kx), the substitution method is used to simplify the integral. This involves substituting u = kx, which adjusts the differential dx accordingly, resulting in the necessary correction factor in the final integrated expression.

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