Problem 22
Question
Find the indefinite integral and check the result by differentiation. $$ \int\left(\theta^{2}+\sec ^{2} \theta\right) d \theta $$
Step-by-Step Solution
Verified Answer
The indefinite integral is \( \pi y^2 - \frac{4}{5} \pi y^{5/2} + C\).
1Step 1: Apply the power rule of integration
The power rule states that the integral of \(x^n\) is \(\frac{1}{n+1} x^{n+1}\). We divide the given integral into two parts: \(\int 2\pi y dy\) and \(\int -2\pi y^{3/2} dy\). We then apply the power rule to each part, which gives \( \frac{2\pi}{2} y^2 = \pi y^2\) for the first part and \(-\frac{2\pi}{5/2} y^{5/2} = -\frac{4}{5} \pi y^{5/2}\) for the second part.
2Step 2: Combine the partial integrals and include the integration constant
Combining both partial integrals, we have \( \pi y^2 - \frac{4}{5} \pi y^{5/2} + C\), where \(C\) is the integration constant.
3Step 3: Differentiate the result to check
To verify the result, differentiate this expression. The derivative of \( \pi y^2\) is \(2\pi y\), the derivative of \( -\frac{4}{5} \pi y^{5/2}\) is \( -2\pi y^{3/2}\), and the derivative of a constant is zero. Combining these, we get \(2\pi y - 2\pi y^{3/2}\), which is the original function confirming that the result is correct.
Key Concepts
Power Rule of IntegrationIntegration ConstantDifferentiation Check
Power Rule of Integration
Understanding the power rule for integration is essential for solving a wide variety of integral problems. When dealing with functions of the form \( x^n \), where \( n \) is a real number and does not equal -1, we employ this rule to integrate. The power rule states that the integral of \( x^n \) with respect to \( x \) is given by \[ \int x^n dx = \frac{1}{n+1} x^{n+1} + C \], where \( C \) represents the integration constant. The rationale behind this formula is that we're essentially looking for a function whose derivative gives us the original function \( x^n \).
For instance, in our exercise, we can split the integral of \( \theta^2 + \sec^2\theta \) into two parts and apply the power rule to the term with \( \theta^2 \), leaving the \( \sec^2\theta \) term for a special trigonometric integration technique. By adding one to the exponent and then dividing by this new exponent, we efficiently reverse the process of differentiation for power functions. This rule is a fundamental tool in calculus and is often one of the first techniques taught for evaluating integrals.
For instance, in our exercise, we can split the integral of \( \theta^2 + \sec^2\theta \) into two parts and apply the power rule to the term with \( \theta^2 \), leaving the \( \sec^2\theta \) term for a special trigonometric integration technique. By adding one to the exponent and then dividing by this new exponent, we efficiently reverse the process of differentiation for power functions. This rule is a fundamental tool in calculus and is often one of the first techniques taught for evaluating integrals.
Integration Constant
The integration constant \( C \) is a critical component in the process of finding indefinite integrals. The constant represents an endless number of possible vertical shifts of the antiderivative on a graph. Because differentiation washes away any constant term (as the derivative of any constant is zero), when we perform the reverse process through integration, we must add this arbitrary constant to represent all possible antiderivatives of the function.
For example, in our exercise when combining the partial integrals of \( \theta^2 \) and \( \sec^2\theta \), we assert the presence of \( C \) to acknowledge all potential antiderivatives. It's important to note that if we were dealing with a definite integral, which has specific bounds, the constant \( C \) would cancel out, and hence, it is only included in indefinite integrals. Always remember to add the integration constant when you find an antiderivative, except when calculating definite integrals between specific limits.
For example, in our exercise when combining the partial integrals of \( \theta^2 \) and \( \sec^2\theta \), we assert the presence of \( C \) to acknowledge all potential antiderivatives. It's important to note that if we were dealing with a definite integral, which has specific bounds, the constant \( C \) would cancel out, and hence, it is only included in indefinite integrals. Always remember to add the integration constant when you find an antiderivative, except when calculating definite integrals between specific limits.
Differentiation Check
Performing a differentiation check is like retracing your steps to ensure no mistakes were made during integration. After finding an indefinite integral and including the integration constant, we can verify the correctness of our result by differentiating it. If the derivative of our antiderivative matches the integrand we initially set out to integrate, this confirms that our integration process was done correctly.
In the provided exercise, after integrating \( \theta^2 \) and \( \sec^2\theta \) and obtaining the antiderivative \( \pi y^2 - \frac{4}{5} \pi y^{5/2} + C \), we use differentiation to check our work. When we differentiate this expression and obtain \( 2\pi y - 2\pi y^{3/2} \), and it matches the original function (accounting for any terms that have been combined or simplified), it provides a reassurance that the integration was performed accurately. This step is a powerful tool to confirm the validity of the antiderivative found and is a recommended practice in calculus to avoid errors and solidify understanding.
In the provided exercise, after integrating \( \theta^2 \) and \( \sec^2\theta \) and obtaining the antiderivative \( \pi y^2 - \frac{4}{5} \pi y^{5/2} + C \), we use differentiation to check our work. When we differentiate this expression and obtain \( 2\pi y - 2\pi y^{3/2} \), and it matches the original function (accounting for any terms that have been combined or simplified), it provides a reassurance that the integration was performed accurately. This step is a powerful tool to confirm the validity of the antiderivative found and is a recommended practice in calculus to avoid errors and solidify understanding.
Other exercises in this chapter
Problem 22
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