Problem 1 - Calculus

Question

Find the indefinite integrels in \(1-20\). \(\int \sqrt{2+x} d x \quad(a d d+C)\)

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{2(2+x)^{3/2}}{3} + C \).

Step 1: Identify the Type of Integral

This integral involves the square root of an expression in the form of \( \sqrt{2+x} \). The goal is to integrate this with respect to \( x \) and add a constant of integration, \( C \).

Step 2: Substitution

Let's use the substitution method to simplify the integral. Let \( u = 2 + x \). Then, \( du = dx \). Substitute these into the integral to get \( \int \sqrt{u} \, du \).

Step 3: Applying Power Rule for Integration

Using the power rule for integration, which states that \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \) for \( n eq -1 \), we integrate \( u^{1/2} \) to get \( \frac{u^{3/2}}{3/2} + C \).

Step 4: Simplify the Result

Simplify \( \frac{u^{3/2}}{3/2} + C \) into \( \frac{2u^{3/2}}{3} + C \).

Step 5: Substitute Back the Original Variable

Return to the original variable by substituting \( u = 2 + x \) back into the expression to get \( \frac{2(2+x)^{3/2}}{3} + C \).

Key Concepts

Substitution Method

The substitution method is a handy tool when dealing with functions that seem complicated at first glance. It transforms an integral into a simpler form, making the integration process more manageable. In this specific exercise, we applied the substitution method to the integral \( \int \sqrt{2+x} \, dx \). Here's how it works step-by-step:

• First, identify a substitution that will simplify the expression. Often, this "substitute" is the inside part of a composite function. For instance, if your function contains \( \sqrt{2+x} \), a useful substitution would be \( u = 2 + x \).
• Next, compute the derivative of the substitution, \( du \). For this example, since \( u = 2 + x \), \( du = dx \).
• Now, replace all instances of \( x \) in the integral with \( u \) and \( du \). This changes our integral from \( \int \sqrt{2+x} \, dx \) to \( \int \sqrt{u} \, du \).

By simplifying the integrand, the integral becomes easier to solve. After integration, always remember to substitute back the original variable.

Power Rule for Integration

The power rule for integration is a fundamental technique in calculus used to integrate power functions. It states that for any function of the form \( u^n \), where \( n eq -1 \), the integral is given by:\[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \]In our exercise, after substitution, we were left with the integral \( \int \sqrt{u} \, du \). To apply the power rule here:

• Notice that \( \sqrt{u} \) can be rewritten as \( u^{1/2} \).
• Using the power rule, integrate \( u^{1/2} \), giving \( \frac{u^{3/2}}{3/2} + C \).
• Simplify this result to \( \frac{2u^{3/2}}{3} + C \).

Applying the power rule correctly is crucial in determining the antiderivative of power functions, allowing for more complex integral calculations.

Constant of Integration

The concept of a constant of integration, represented as \( C \), is central to finding indefinite integrals. Whenever we integrate a function without given limits, we introduce this constant.An indefinite integral represents a family of functions, all of which differ by a constant value. Why include it?

• Integration essentially reverses differentiation; however, differentiating different constants (like \( C \)) results in zero. This implies that many different functions can have the same derivative, i.e., the initial function before difference.
• By including \( C \), we ensure that we account for any possible constant that could have been there originally before differentiation.

In our solution for \( \int \sqrt{2+x} \, dx \), \( C \) was added at the end, yielding the expression \( \frac{2(2+x)^{3/2}}{3} + C \). Always remember: the constant of integration is necessary to represent all potential antiderivatives of a function.