Problem 43 - Calculus Single Variable

Question

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

Step-by-Step Solution

Answer

The integral is \( \frac{1}{\ln(4)} 4^x + C \).

Step 1: Identify the Integral Form

The given integral is \( \int 2^{x \ln (2)} \, dx \). The expression \( 2^{x \ln (2)} \) can be rewritten using exponent properties as \( e^{x \ln (2^2)} = e^{2x \ln (2)} \). Thus, the integral is \( \int e^{2x \ln(2)} \, dx \).

Step 2: Simplify the Exponent

By recognizing that \( 2^2 = 4 \), the integral expression can also be rewritten as \( \int e^{x \ln (4)} \, dx \). This makes it easier by expressing the exponentiation in terms of the natural exponential function.

Step 3: Use a Substitution

Let \( u = x \ln(4) \). Then, \( \frac{du}{dx} = \ln(4) \), or \( dx = \frac{du}{\ln(4)} \). Therefore, the integral becomes \( \int e^u \cdot \frac{1}{\ln(4)} \, du \).

Step 4: Integrate

The integral \( \int e^u \, du \) is \( e^u + C \). Substituting back \( u = x \ln(4) \), we find \( \frac{1}{\ln(4)} \int e^u \, du = \frac{1}{\ln(4)} e^{x \ln(4)} + C \).

Step 5: Substitute Back

Since \( e^{x \ln(4)} = 4^x \), the integral simplifies to \( \frac{1}{\ln(4)} 4^x + C \). Therefore, the indefinite integral \( \int 2^{x \ln(2)} \, dx \) is \( \frac{1}{\ln(4)} 4^x + C \).

Key Concepts

Exponent Properties

Exponent properties are the rules that govern how exponents behave in expressions. These properties help simplify expressions and solve equations involving exponents.
One key property is the "power of a power" rule. It states that when raising an exponent to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\). This property appears in our integral solution, where we handle \(2^{x \ln(2)}\).

• If we have an expression like \(a^{b+c}\), it can be split into \(a^b \cdot a^c\).
• An expression like \(a^{-b}\) is equivalent to \(\frac{1}{a^b}\), helping to manage negative exponents.
• The property \((ab)^n = a^n \cdot b^n\) splits a power distribution over a product.

Understanding these rules lets you rearrange and simplify expressions for calculus solutions, like recognizing that \(2^{x \ln(2)}\) can be shifted to an exponential form easier to integrate.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for any value of the variable. They are extremely useful in calculus for simplifying integrals involving trigonometric functions.
Most commonly used are fundamental identities like the Pythagorean identities:

• \(\sin^2(x) + \cos^2(x) = 1\)
• \(1 + \tan^2(x) = \sec^2(x)\)
• \(1 + \cot^2(x) = \csc^2(x)\)

Other useful trigonometric identities include the angle sum and difference identities:

• \(\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b\)
• \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)

While the immediate integral in the exercise wasn't directly involving trigonometric functions, these identities often assist in transforming integrals into more manageable forms. Proper understanding allows complex trigonometric integrals to be solved more straightforwardly.

Substitution Method

The substitution method, also known as \(u\)-substitution, is a technique in calculus to simplify an integral by substituting part of the integral with a new variable. This method is particularly useful when the integral involves a composition of functions.
To use the substitution method, follow these steps:

• Identify a part of the integral to substitute. Pick \(u\), a function inside the integral that simplifies it.
• Differentiate \(u\) to find \(\frac{du}{dx}\) and solve for \(dx\).
• Replace the identified part of the integral and \(dx\) with \(du\) and the corresponding expression.
• Integrate with respect to \(u\) and then substitute back the original variable.

In the exercise provided, we let \(u = x \ln(4)\), and this transformed the integral into a more straightforward exponential integral. By substituting back, the problem was simplified, demonstrating the power of substitution to make complicated integrals more manageable.