Question

In Problems \(1-16\), perform the indicated integrations. \(\int x \sqrt{x+1} d x\)

Short answer

The integral is \( \frac{2}{5}(x+1)^{5/2} - \frac{2}{3}(x+1)^{3/2} + C \).

Step-by-step solution

Identify the Type of Integration

The integral \( \int x \sqrt{x+1} \, dx \) involves a product of a polynomial \(x\) and a square root \(\sqrt{x+1}\). This often indicates that a substitution method may simplify the integral.

Choose a Substitution

Let \( u = x + 1 \), which implies \( du = dx \). Consequently, \( x = u - 1 \). Substitute these into the integral.

Transform the Integral

Substitute \( x = u - 1 \) and \( dx = du \) into the integral: \[ \int x \sqrt{x+1} \, dx = \int (u-1) \sqrt{u} \, du = \int (u-1)u^{1/2} \, du \] Expand this to \[ \int (u^{3/2} - u^{1/2}) \, du \]

Integrate Term by Term

Integrate each term separately:- For \( \int u^{3/2} \, du \), use the power rule \( \frac{u^{5/2}}{5/2} = \frac{2}{5} u^{5/2} \)- For \( \int u^{1/2} \, du \), use the power rule \( \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2} \)Thus, the integral becomes \[ \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2} + C \]

Substitute Back to Original Variable

Replace \( u \) with \( x + 1 \):\[ \frac{2}{5} (x+1)^{5/2} - \frac{2}{3} (x+1)^{3/2} + C \]

Verify the Integration

Differentiate the result to ensure it simplifies back to the original integrand \( x \sqrt{x+1} \), confirming the correctness of the integral.

Key concepts

Substitution Method

The substitution method is a powerful integration technique used to simplify complex integrals by changing variables. In our example, the given integral is \( \int x \sqrt{x+1} \, dx \). Often, when you see a composite function such as a square root, a substitution can make the integration process easier.

• Start by identifying a part of the integrand that can be replaced with a simpler expression.
• In this case, letting \( u = x + 1 \) simplifies the square root \( \sqrt{x+1} \) to \( \sqrt{u} \).
• Then, calculate \( du = dx \), and rearrange \( x \) as \( x = u - 1 \).

Substituting these into the integral transforms it to \( \int (u-1) \sqrt{u} \, du \), which is much simpler to evaluate. The substitution helps break down a potentially difficult integral into manageable parts, by transforming the complex expression into a basic polynomial expression.

Power Rule

The power rule for integration is an essential technique that allows you to integrate polynomials easily. This rule states that \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\] if \( n eq -1 \), where \( C \) is the constant of integration.In our problem, after substitution, the integral becomes \( \int (u^{3/2} - u^{1/2}) \, du \). Applying the power rule separately to each term:

• For \( \int u^{3/2} \, du \), we use the power rule to get \( \frac{u^{5/2}}{5/2} = \frac{2}{5} u^{5/2} \).
• For \( \int u^{1/2} \, du \), apply the power rule to obtain \( \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2} \).

Thus, the original power expressions have been simplified to a solved integral by applying the power rule to each polynomial individually. Remembering this rule can greatly aid in quickly integrating polynomial terms!

Integrals of Polynomials

When dealing with integrals of polynomials, the process becomes straightforward with the application of the power rule. In the integration process of our example, once the substitution \( u = x + 1 \) was made, we expanded the polynomial expression \[(u-1)u^{1/2} = u^{3/2} - u^{1/2}\] and introduced separate terms.

• Each resulting term \((u^{3/2})\), \(( - u^{1/2})\) was a polynomial and ready to be integrated using the power rule.

Integrating polynomials involves increasing the power of each term by one and dividing by the new power. This consistent approach is what makes polynomial integrals easy to handle, as they follow predictable patterns and yield straightforward results.

Verification of Integration

Verification of integration is the final step where we confirm our result by differentiating it, ensuring it returns to the original integrand. In this exercise, we derived the integrated function \[\frac{2}{5} (x+1)^{5/2} - \frac{2}{3} (x+1)^{3/2} + C\]To verify:

• Differentiate the function with respect to \( x \).
• After applying the chain rule appropriately, the resulting function should match the original function \( x \sqrt{x+1} \).
• Ensure all transformations back to the variable \( x \) are correct, paralleling the initial substitution.

By following these checks, verification acts as a quality control measure, confirming that the integration was executed accurately and reflecting the elegance of calculus in turning functions back into their integrands.