Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int \sqrt{x} e^{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The evaluation of the integral \(\int \sqrt{x} e^{\sqrt{x}} d x\) is \(2x e^{\sqrt{x}} - 4\sqrt{x}e^{\sqrt{x}} + 4e^{\sqrt{x}} + C\).

Step 1: Apply The Substitution

Firstly perform the suggested substitution \(w=\sqrt{x}\), which implies that \(w^{2}=x\). Differentiate both sides of the equation \(w^{2}=x\) to get, \(2wdw=dx\). The integral becomes \(\int w e^{w} 2w d w\).

Step 2: Simplify The Integral

After applying the substitution, the integral can be simplified to this new form: \(\int 2w^{2} e^{w} d w\).

Step 3: Apply Integration By Parts

Next, use a second method - the method of integration by parts. For this, take \(u = 2w^2\), which implies that \(du = 4wdw\) and \(dv = e^{w} dw\) which implies \(v = e^{w}\). With these new substitutions, apply the formula for integration by parts which states: \(\int u dv = uv - \int v du\). Upon substituting, the integral becomes \(2w^2 e^{w} - \int 4wdx * e^{w}\).

Step 4: Evaluate The Integral

Now, integrate what is left: \(2w^2 e^{w} - \int 4wdx * e^{w} = 2w^2 e^{w} - 4 \int wde^{w}\). We can apply the method of integration by parts again. This time take \(u = w\), so \(du = dw\), \(dv = e^{w} dw\), so \(v = e^{w}\). After substituting the integral becomes \(2w^2 e^{w} - 4(we^{w} - \int e^{w})\). This integral is now simpler and can be integrated directly to be \(2w^2 e^{w} - 4we^{w} + 4e^{w}\).

Step 5: Back-substitute The Original Variable

Finally re-substitute the variable \(w\) with \(\sqrt{x}\) which was the original substitution. So the solution to the integral in the original terms is \(2x e^{\sqrt{x}} - 4\sqrt{x}e^{\sqrt{x}} + 4e^{\sqrt{x}} + C\), where \(C\) is the constant of integration.

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus integration. It simplifies complex integrals by replacing a variable with another expression. This is helpful when dealing with difficult functions that involve nested functions or complex compositions. In our exercise, the substitution was:

• Set \( w = \sqrt{x} \) which simplifies the process.
• This leads to \( w^2 = x \) making \( 2wdw = dx \), allowing us to replace the entire integral.

By using the substitution \( w = \sqrt{x} \), the expression under the integral sign transformed into a form that was easier to manage. This allows us to deal with a simpler function, ultimately making the task of integration far more straightforward.

Integration by Parts

Integration by parts is a valuable method for solving integrals that involve the product of two functions, based on the formula \( \int u\, dv = uv - \int v\, du \). It's the integration counterpart of the product rule in differentiation. Here's how it worked in our example:

• Select \( u = 2w^2 \) and \( dv = e^w \, dw \).
• This leads to \( du = 4w\, dw \) and \( v = e^w \).

After substituting these into the integration by parts formula, the original problem transformed and became more approachable. It involved repetitive application of integration by parts and achieving a simpler form for evaluation. The key is choosing \( u \) such that \( du \) simplifies the process.

Differential Equations

Differential equations may seem a step aside from standard integration at first, but they are inherently connected. While this particular problem does not directly involve differential equations, understanding the derivatives and integrals of parts we manipulate helps build a bridge between the two. Fundamental concepts like these bridge single integrals with larger systems of equations seen in differential equations.

• Remember, derivatives form the basis of these equations, similar to how they are utilized in integration techniques.
• Learning to manage these connected parts efficiently is vital for solving complex mathematical models.

Consider differential equations as the big picture where integrations are often small but crucial parts.

Indefinite Integrals

Indefinite integrals are integrals without specific limits, representing a family of functions with an arbitrary constant \( C \). After integration is performed, it's crucial to include this constant as part of the final expression. Here's why it's so essential:

• Indicates a family of solutions rather than a single answer.
• Ensures the generality of equations solved via integration, like we saw in our exercise solution.

The constant, \( C \), acknowledges that multiple solutions satisfy an indefinite integral. In our solution, after back-substituting \( w = \sqrt{x} \), the final expression included \( C \) to reflect all possible antiderivatives resulting from the integration process.