Problem 16 - Calculus Early Transcendentals

Question

Use the given substitution to find the following indefinite integrals. Check your answer by differentiating. $$\int(6 x+1) \sqrt{3 x^{2}+x} d x, u=3 x^{2}+x$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the expression \((6x+1)\sqrt{3x^2+x}\) using the substitution \(u=3x^2+x\). Answer: The indefinite integral of the expression \((6x+1)\sqrt{3x^2+x}\) using the substitution \(u=3x^2+x\) is \((3x^2+x)+C\).

Step 1: Calculate du

Given the substitution \(u=3x^2+x\), let's calculate the derivative of \(u\) with respect to \(x\): $$\frac{d u}{d x} = \frac{d}{d x}(3x^2+x) = 6x+1.$$ Now, we can solve the equation $$\mathrm{d}u = (6x+1)\;\mathrm{d}x$$ for \(\mathrm{d}x\): $$\mathrm{d}x = \frac{\mathrm{d}u}{6x+1}.$$

Step 2: Express the integral in terms of \(u\)

Given the integral $$\int (6x+1)\sqrt{3x^2+x}\,\mathrm{d}x,$$ we have to substitute the expression for \(u\) and \(\mathrm{d}x\): Recall that \(u = 3x^2+x\). We can rewrite the integral in terms of \(u\): $$\int \sqrt{u}\frac{\mathrm{d}u}{6x+1}.$$

Step 3: Simplify the integral

Notice that \(\sqrt{u}\) and \(\frac{\mathrm{d}u}{6x+1}\) simplify to 1, so we have: $$\int 1\,\mathrm{d}u.$$

Step 4: Integrate

Now, we integrate with respect to \(u\): $$\int 1\,\mathrm{d}u = u + C,$$ where \(C\) is the constant of integration.

Step 5: Replace \(u\) with the expression in terms of \(x\)

Since we know that \(u = 3x^2+x\), we replace \(u\) with the given expression and obtain the integral in terms of \(x\): $$u + C = (3x^2+x) + C.$$

Step 6: Check the answer by differentiating

Now, we differentiate the obtained expression with respect to \(x\) to verify our answer: $$\frac{d}{dx}((3x^2+x)+C) = 6x+1.$$ Thus, the original integral expression \((6x+1)\sqrt{3x^2+x}\) was successfully integrated, and the result is \((3x^2+x)+C\).

Key Concepts

Substitution Method

One common method for solving indefinite integrals is the substitution method. This technique involves choosing a substitution, usually denoted by a new variable like \(u\). The main goal is to simplify the integral, making it easier to evaluate. In this exercise, we used the substitution \(u = 3x^2 + x\). This substitution replaces the complex expression inside the integral with a single variable.To use substitution, determine the derivative of the chosen substitution with respect to the original variable, in this case \(x\). This derivative informs us how to express \(dx\) in terms of \(du\). Here, \(\frac{du}{dx} = 6x + 1\), resulting in \(dx = \frac{du}{6x+1}\). After substituting into the integral, we transform it into a function of \(u\). This often leads to a simpler integral that is straightforward to solve. After solving the integral with respect to \(u\), we must replace the original substitution to express the final result in terms of the original variable \(x\). It's important to remember to include the constant of integration \(C\) at the end.

Integration Techniques

There are many integration techniques that can help in evaluating indefinite integrals. Each technique serves to simplify the expression being integrated.

• Substitution: Already discussed, it simplifies an integral by reformatting it in terms of a new variable.
• Integration by Parts: Another method involving parts of the function, not used here but useful for products of functions.
• Partial Fraction Decomposition: Used when you have a rational expression, useful for separating complex fractions into simpler parts.

In our exercise, substitution transforms \((6x+1)\sqrt{3x^2 + x} dx\) into a much simpler integral. This method is particularly effective when an expression inside the integral or its derivative is directly present in the integrand, as we saw with the function \(3x^2 + x\). Once we've substituted effectively, we simplify the integral to a form that often results in an easier integration step.

Differentiation

After solving an integral, it's crucial to differentiate the result to verify correctness. Differentiation is the reverse operation of integration. By doing so, you ensure that integrating and differentiating are consistent processes. In this exercise, we derived \(u = 3x^2 + x\) back into the original form. Differentiation of the solution \((3x^2 + x) + C\) should return us to the original integrand \((6x + 1) \sqrt{3x^2 + x}\). As shown, when we differentiated \(3x^2 + x\), it returned \(6x + 1\), confirming that our integration was correct.Differentiation has various rules like the product rule, quotient rule, and chain rule. Here, we simply used the basic power rule, which is usually sufficient for recalculating many standard derivatives. Simplifying checks with differentiation ensures the integrity of calculus work by validating the steps and solution.