Problem 12 - Essential Calculus Early Transcendentals

Question

Find the most general antiderivative of the function (Check your answer by differentiation $$f(x)=2 \sqrt{x}+6 \cos x$$

Step-by-Step Solution

Answer

The most general antiderivative is \( F(x) = \frac{4}{3}x^{3/2} + 6\sin x + C \).

Step 1: Identify the terms in the function

The given function is \( f(x) = 2\sqrt{x} + 6\cos x \). We have two terms to integrate: \( 2\sqrt{x} \) and \( 6\cos x \). Each term will be integrated separately.

Step 2: Integrate the term \(2\sqrt{x}\)

First, rewrite \( \sqrt{x} \) as \( x^{1/2} \). Therefore, \( 2\sqrt{x} = 2x^{1/2} \).To find its integral, apply the power rule for integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \).- Here, \( n = \frac{1}{2} \), so the integral becomes: \( \int 2x^{1/2} dx = 2 \cdot \frac{x^{3/2}}{3/2} = \frac{4}{3}x^{3/2} \).

Step 3: Integrate the term \(6\cos x\)

To integrate \( 6\cos x \), use the cosine integral rule: \( \int \cos x \, dx = \sin x + C \).So, \( \int 6\cos x \, dx = 6 \sin x \).

Step 4: Combine the integrals and add the constant

The general antiderivative of the function will be the sum of the individual integrals, each having a constant \( C \).Thus, the antiderivative is:\[ F(x) = \frac{4}{3}x^{3/2} + 6\sin x + C \]where \( C \) is an arbitrary constant of integration.

Step 5: Verify by Differentiation

Differentiate \( F(x) = \frac{4}{3}x^{3/2} + 6\sin x + C \) to verify:- The derivative of \( \frac{4}{3}x^{3/2} \) using the power rule is \( 2x^{1/2} \).- The derivative of \( 6\sin x \) is \( 6\cos x \).- The derivative of a constant \( C \) is 0.Hence, the derivative \( \frac{d}{dx}F(x) = 2x^{1/2} + 6\cos x = f(x) \), which confirms our antiderivative is correct.

Key Concepts

Integration

Integration is a fundamental concept in calculus that involves finding the antiderivative, or the inverse operation, of differentiation. Essentially, when you integrate a function, you are looking for another function whose derivative would give you the original function. Integration allows us to find areas under curves, compute accumulated quantities, and solve differential equations.

• The Function: In our example, the function is composed of two terms: \( 2\sqrt{x} \) and \( 6\cos x \). These terms represent a sum of a polynomial-like function and a trigonometric function.
• Sum Rule: When a function is the sum of two or more terms, you can integrate each term separately. This property is known as the sum rule of integration.

The goal in integration is to find a general antiderivative, which includes a constant \( C \) because the derivative of any constant is zero, which means that it will disappear upon differentiation.

Power Rule

The power rule for integration is a handy tool for integrating polynomial expressions, such as \( x^{n} \). The rule states that the integral of \( x^{n} \) is \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \).

• Application: In our exercise, we applied the power rule to the term \( 2\sqrt{x} \), first rewriting the square root as a power: \( x^{1/2} \).
• Calculation: Applying the power rule, we found \( \int 2x^{1/2} \, dx \). For this, we incremented the power of \( x \) by one, resulting in \( x^{3/2} \), and then divided by the new power: \( \frac{4}{3}x^{3/2} \).

This straightforward approach makes it simple to handle terms with power functions, facilitating the integration process for polynomials.

Trigonometric Integrals

Trigonometric integrals involve integrating functions composed of trigonometric functions like sine, cosine, tangent, etc. These functions have specific integral formulas that are useful in various applications.

• Cosine Integral: In our problem, we needed to integrate \( 6\cos x \). We used the integral rule for cosine, \( \int \cos x \, dx = \sin x + C \).
• Solution: Applying this rule, the integral of \( 6\cos x \) becomes \( 6\sin x \), incorporating the coefficient in the result.

Understanding these integral rules helps in solving integrals quickly and effectively, especially within more complex expressions containing trigonometric functions.