Question

Variations on the substitution method Evaluate the following integrals. $$\int(z+1) \sqrt{3 z+2} d z$$

Short answer

Question: Evaluate the integral of the function \((z + 1) \sqrt{3z + 2}\). Answer: \(\frac{1}{9} \left[\frac{2}{5} (3z + 2)^{\frac{5}{2}} - \frac{2}{3} (3z + 2)^{\frac{3}{2}} \right] + C\)

Step-by-step solution

Choose the substitution

Choose the substitution \(u=3z+2\).

Find the derivative of the substitution

Find the derivative of \(u\) with respect to \(z\), which is \(du = 3 dz\).

Rewrite the integral in terms of u

Divide both sides of \(du = 3 dz\) by 3 and isolate \(dz\) and we get \(dz = \frac{1}{3} du\). Substitute \(dz\) and \(u = 3z + 2\) into the integral: $$ \int (z+1) \sqrt{3z+2} dz = \int \frac{u - 1}{3} \sqrt{u} \cdot \frac{1}{3} du $$

Simplify the integrand and integrate

Now, simplify the integrand by multiplying the constants and combining the terms as follows: $$ \frac{1}{9} \int (u - 1) \sqrt{u} du $$ Now, let's expand and evaluate the integral: $$ \frac{1}{9} \int (u\sqrt{u} - \sqrt{u}) du = \frac{1}{9} \int u^{\frac{3}{2}} du - \frac{1}{9} \int u^{\frac{1}{2}} du $$ Next, apply the power rule on both integrals: $$ \frac{1}{9} \left[\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}} \right] + C $$

Substitute back z

Now, substitute back \(z\) by using the substitution \(u=3z+2\): $$ \frac{1}{9} \left[\frac{2}{5} (3z + 2)^{\frac{5}{2}} - \frac{2}{3} (3z + 2)^{\frac{3}{2}} \right] + C $$ This is the final evaluated integral.

Key concepts

Substitution Method

The substitution method is a powerful integration technique that helps simplify complex integrals. It involves substituting a portion of the original integrand with a new variable, making the integral easier to evaluate.
In this exercise, we used the substitution \(u = 3z + 2\). This choice simplifies the integral by transforming it into a form that is simpler to integrate. When you choose a substitution, it's vital to express all parts of the integrand, including \(dz\), in terms of the new variable \(u\).

• First, find the derivative of the substitution: \(du = 3 \, dz\).
• Then, express \(dz\) in terms of \(du\): \(dz = \frac{1}{3} \, du\).
• Substitute these expressions back into the integral to transform it.

This method effectively reduces the complexity, allowing us to apply straightforward integration techniques such as the power rule later.

Definite Integral

While the original exercise involves an indefinite integral, understanding definite integrals is crucial as well. A definite integral is used to calculate the area under a curve between two specified limits.
It is expressed in the form \(\int_{a}^{b} f(x) \, dx\), where \(a\) and \(b\) are the limits of integration. The definite integral evaluates to a number that represents this area.

• The substitution method transforms the limits along with the variable. If \(z = a\), then \(u = 3a + 2\), and similarly for \(z = b\).
• This ensures that the integral's evaluation remains consistent within the new bounds.
• Finally, after finding the antiderivative, apply the fundamental theorem of calculus to calculate the definite integral within these limits.

Definite integrals are fundamental in solving real-world problems, ranging from physics to engineering, where precise quantities need calculation over specific intervals.

Power Rule

The power rule is a fundamental technique in calculus used to find the antiderivative of basic power functions. It states that for any real number \(n eq -1\), the integral of \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\).
In the context of the given integral, the power rule is applied after simplifying the integrand using substitution.

• For \(\int u^{\frac{3}{2}} du\), the antiderivative is \(\frac{2}{5} u^{\frac{5}{2}}\).
• For \(\int u^{\frac{1}{2}} du\), the antiderivative is \(\frac{2}{3} u^{\frac{3}{2}}\).
• These particular transformations are executed by applying the power rule separately to each term.

This rule allows us to solve the integral by breaking it into more manageable parts. Always remember to revert to the original variable after integrating when using substitution to ensure the solution accurately reflects the initial integral expression.