Problem 44 - Calculus Single Variable

Question

Use trigonometric identities to compute the indefinite integrals. Evaluate \(\int \frac{10^{x}}{3^{2 x}} d x\)

Step-by-Step Solution

Answer

The indefinite integral evaluates to \( \frac{\left( \frac{10}{9} \right)^x}{\ln \left( \frac{10}{9} \right)} + C \).

Step 1: Simplify the Expression

The original integral is \(\int \frac{10^x}{3^{2x}} \, dx\). We can rewrite the integral as \(\int 10^x \cdot \frac{1}{3^{2x}} \, dx\), which simplifies to \(\int \left( \frac{10}{9} \right)^x \, dx\). This simplification helps in recognizing the form of the integrand.

Step 2: Use the Exponential Integral Formula

Recall the formula for the integral of an exponential function: \(\int a^x \, dx = \frac{a^x}{\ln a} + C\), where \(a > 0\). Here, \(a = \frac{10}{9}\). Substitute this into the formula: \(\int \left( \frac{10}{9} \right)^x \, dx = \frac{\left( \frac{10}{9} \right)^x}{\ln \left( \frac{10}{9} \right)} + C\).

Key Concepts

Trigonometric Identities

Trigonometric identities are critical tools in calculus, especially when integrating complex trigonometric functions. Although the given exercise does not explicitly involve trigonometric functions, understanding how to use identities can greatly simplify integration processes in other contexts.

Trigonometric identities include formulas like the Pythagorean identity:

• Sin²θ + Cos²θ = 1
• Angle sum and difference identities, such as: Sin(A ± B) = Sin(A)Cos(B) ± Cos(A)Sin(B)

These formulas allow transformation of complex expressions into simpler forms.

In integration, this can translate into converting products or compositions of trigonometric functions into basic forms that are easier to integrate. Mastery of trigonometric identities eases handling of a wide range of calculus problems.

Exponential Integral Formula

The exponential integral formula is a powerful tool for finding indefinite integrals involving exponential functions. In this exercise, the integral \[ \int \left( \frac{10}{9} \right)^x \ dx \] is simplified using this formula.

The general exponential integral formula is:

• \( \int a^x \, dx = \frac{a^x}{\ln a} + C \)

where \( a > 0 \) and \( a eq 1 \). This formula directly applies when the function is of the exponential form \( a^x \).

In our exercise, it transforms the complex exponential integral into a basic computation involving logarithms, specifically: \[ \int \left( \frac{10}{9} \right)^x \ dx = \frac{\left( \frac{10}{9} \right)^x}{\ln \left( \frac{10}{9} \right)} + C \] Understanding this formula is essential because exponential functions appear frequently in calculus. Recognizing when to use it simplifies calculations and leads to quick solutions.

Simplifying Expressions

Simplifying expressions is an important step in solving many calculus problems, especially when dealing with integrals. The original integral in this exercise was \[ \int \frac{10^x}{3^{2x}} \, dx \] By transforming it into \[ \int \left( \frac{10}{9} \right)^x \, dx \] we made the problem much easier to handle by focusing on its exponential nature.

Simplification often involves:

• Breaking down complex fractions into smaller components
• Factoring and reducing powers where possible
• Rewriting products of terms into single exponential terms

Here, simplifying \( \frac{10^x}{3^{2x}} \) into \( \left( \frac{10}{9} \right)^x \) closed the gap between recognizing the integral's form and applying the exponential integral formula.

This step is crucial because it transforms the integrand into a format where established integration rules can be easily applied, reducing the complexity of the problem.