Problem 24
Question
Evaluate the indicated indefinite integrals. $$ \int(z+\sqrt{2} z)^{2} d z $$
Step-by-Step Solution
Verified Answer
\[\frac{(3 + 2\sqrt{2})z^3}{3} + C\]
1Step 1: Simplify the Integrand
The integrand is \((z + \sqrt{2} z)^2\). Factor out \(z\) from the parentheses: \(z(1 + \sqrt{2})\). Expanding the square gives: \([z(1+\sqrt{2})]^2 = z^2(1 + \sqrt{2})^2 = z^2(1 + 2\sqrt{2} + 2) = z^2(3 + 2\sqrt{2})\). The integrand simplifies to \((3 + 2\sqrt{2})z^2\).
2Step 2: Integrate the Simplified Function
Integrate the simplified function \((3 + 2\sqrt{2})z^2\). The integral becomes \((3 + 2\sqrt{2})\int z^2 \, dz\). Use the power rule for integration, which states \(\int z^n \, dz = \frac{z^{n+1}}{n+1} + C\). Here, \(n = 2\), so \(\int z^2 \, dz = \frac{z^3}{3}\).
3Step 3: Apply the Constant Multiplier
Multiply the integrated expression by the constant multiplier \((3 + 2\sqrt{2})\). The indefinite integral becomes \((3 + 2\sqrt{2}) \cdot \frac{z^3}{3} = \frac{(3 + 2\sqrt{2})z^3}{3}\).
4Step 4: Add the Constant of Integration
Since this is an indefinite integral, we need to add the constant of integration \(C\) at the end. The final result of the integral is \[\frac{(3 + 2\sqrt{2})z^3}{3} + C\].
Key Concepts
Power Rule for IntegrationIntegration TechniquesConstant of Integration
Power Rule for Integration
The power rule for integration is a fundamental technique in calculus, especially useful for polynomial functions. When we talk about the power rule, we refer to a straightforward method to integrate functions of the form \(z^n\). This rule simplifies the integration process for students and mathematicians alike.
The core idea of the power rule is that if you have a function \(z^n\), the integral is given by \(\int z^n \, dz = \frac{z^{n+1}}{n+1} + C\), where \(n\) is any real number, except \(-1\). The \(+ C\) represents the constant of integration.
This method stems from reversing the process of differentiation. For instance, when differentiating \(z^{n+1}\), we get back \(n \cdot z^n\). The power rule does the opposite, allowing us to reconstruct the initial function from its derivative. It's important to note that \(C\) is crucial, as it accounts for any constant that could have been in the original function but disappeared during differentiation.
To illustrate, consider our example \(\int z^2 \, dz\). Using the power rule, we increment the power of \(z\) by one, obtaining \(\frac{z^3}{3} + C\), where \(C\) is the constant of integration.
The core idea of the power rule is that if you have a function \(z^n\), the integral is given by \(\int z^n \, dz = \frac{z^{n+1}}{n+1} + C\), where \(n\) is any real number, except \(-1\). The \(+ C\) represents the constant of integration.
This method stems from reversing the process of differentiation. For instance, when differentiating \(z^{n+1}\), we get back \(n \cdot z^n\). The power rule does the opposite, allowing us to reconstruct the initial function from its derivative. It's important to note that \(C\) is crucial, as it accounts for any constant that could have been in the original function but disappeared during differentiation.
To illustrate, consider our example \(\int z^2 \, dz\). Using the power rule, we increment the power of \(z\) by one, obtaining \(\frac{z^3}{3} + C\), where \(C\) is the constant of integration.
Integration Techniques
Integration techniques are strategies or methods employed to find the antiderivative of functions, which is essential in solving integrals. Given the formula \((z+\sqrt{2} z)^2\), it isn't straightforward to apply integration directly. Therefore, simplifying the expression is an effective strategy.
One common technique is **substitution** or **algebraic manipulation**. In our problem, this step involved expanding \((z(1+\sqrt{2}))^2\) to a simpler form, \((3 + 2\sqrt{2})z^2\). This transformation allows us to apply the power rule more efficiently.
**Factorization** is another technique helpful in simplifying integrals. Factoring \(z\) out of \((z+\sqrt{2} z)\) helped in rearranging the expression for easier expansion.
Once simplified, finding the integral becomes a matter of applying rules like the power rule or others if necessary. Students can combine and practice various techniques to become adept at tackling more complex integrals.
One common technique is **substitution** or **algebraic manipulation**. In our problem, this step involved expanding \((z(1+\sqrt{2}))^2\) to a simpler form, \((3 + 2\sqrt{2})z^2\). This transformation allows us to apply the power rule more efficiently.
**Factorization** is another technique helpful in simplifying integrals. Factoring \(z\) out of \((z+\sqrt{2} z)\) helped in rearranging the expression for easier expansion.
Once simplified, finding the integral becomes a matter of applying rules like the power rule or others if necessary. Students can combine and practice various techniques to become adept at tackling more complex integrals.
Constant of Integration
The constant of integration, often denoted by \(C\), is an integral part of indefinite integrals. It emerges from the fact that differentiation of any constant value yields zero.
Thus, when we integrate, we include \(C\) to represent any possible constant that might have been removed during differentiation. It's crucial since without \(C\), the result represents just a family of solutions rather than the general solution.
In practical terms, when solving indefinite integrals like \(\int z^2 \, dz\), even after applying the power rule, we must add \(+ C\). This inclusion acknowledges that there may be multiple valid antiderivatives differing by a constant factor.
Ignoring \(C\) could lead to errors in understanding or applying the integral in further mathematical modeling or real-world applications. Therefore, it's always essential to add the constant of integration unless you're working with definite integrals, where limits are provided.
Thus, when we integrate, we include \(C\) to represent any possible constant that might have been removed during differentiation. It's crucial since without \(C\), the result represents just a family of solutions rather than the general solution.
In practical terms, when solving indefinite integrals like \(\int z^2 \, dz\), even after applying the power rule, we must add \(+ C\). This inclusion acknowledges that there may be multiple valid antiderivatives differing by a constant factor.
Ignoring \(C\) could lead to errors in understanding or applying the integral in further mathematical modeling or real-world applications. Therefore, it's always essential to add the constant of integration unless you're working with definite integrals, where limits are provided.
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