Question

Determine the integrals. $\int 3 \sqrt{10^{3 x}} d x$

   Determine the integrals.
 $\int 3 \sqrt{10^{3 x}} d x$

Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences

Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences

Jr, Ernest F… 1st Edition

Chapter 14, Problem 39

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Recall that $\sqrt{a} = a^{1/2}$. Therefore, $\sqrt{10^{3x}} = (10^{3x})^{1/2}$. Show more…

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Determine the integrals. $\int 3 \sqrt{10^{3 x}} d x$

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Key Concepts

Exponential Functions

Exponential functions are functions of the form a^(kx) where the variable x appears in the exponent. Their distinctive growth properties make them common in many areas of mathematics, and understanding how to work with these functions is essential for solving many integrals and differential equations.

Integration of Exponential Functions

Integrating exponential functions generally involves recognizing the integral of a^(kx) dx, which can often be solved by applying a formula that includes the natural logarithm of the base, such as a^(kx)/(k ln(a)). This technique is essential when the variable appears in the exponent, as it allows one to integrate by normal means.

Substitution Method in Integration

The substitution method is a fundamental technique in integration used to simplify integrals by changing the variable. This method is particularly useful for exponential functions when appropriate substitution transforms the integrand into a standard form that can be directly integrated.

Properties of Exponents

The rules of exponent manipulation, such as rewriting expressions like the square root or power of an exponent, are crucial for simplifying exponential functions. For instance, converting a radical expression into an exponential form can ease the process of integration, revealing more straightforward forms such as a^(kx).

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