Problem 21
Question
Find the indefinite integral and check the result by differentiation. $$ \int(1-\csc t \cot t) d t $$
Step-by-Step Solution
Verified Answer
The indefinite integral of \( t^{2}(t-\frac{2}{t}) dt \) is \( \frac{t^4}{4} - 2t + C \)
1Step 1: Simplification of Integral's Function
Begin by expanding \( t^2 \) across \( (t - \frac{2}{t}) \), which results in the function \( t^3 - 2 \). Now the integral is: \(\int (t^3 - 2) dt \)
2Step 2: Compute the Integral
Integrate term by term using the power rule of integration. The integral for \( t^3 \) becomes \( \frac{t^4}{4} \) and the integral for -2 becomes -2t, as we take the integral of a constant times a variable as the product of the constant and the variable itself. Therefore the function after integration is \( \frac{t^4}{4} - 2t + C \), where C is the constant of integration.
3Step 3: Verify the Integral by Differentiation
Differentiate the result to check if it corresponds to the original integrand. The derivative of \( \frac{t^4}{4} \) is \( t^3 \), the derivative of -2t is -2, and the derivative of a constant \( C \) is 0. Therefore, the derivative of the result of integration is \( t^3 - 2 \), which matches the original function we wanted to integrate. This confirms the correctness of the integration.
Key Concepts
Integration by Power RuleIntegral SimplificationVerification by Differentiation
Integration by Power Rule
Understanding the power rule for integration is essential for solving indefinite integrals involving polynomial expressions. Similar to its counterpart in differentiation, the power rule in integration allows us to easily integrate monomial terms such as \( t^n \), where \( n \) is a real number.
The general form of the power rule for indefinite integrals is \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \) for all values of \( n \) not equal to -1. Here, \( C \) represents the constant of integration, which is added because indefinite integrals represent a family of functions.
In our exercise, when we apply the power rule to the term \( t^3 \), we raise the exponent by 1, resulting in \( t^4 \) and divide by the new exponent, yielding \( \frac{t^4}{4} \) as part of our indefinite integral. The integration of the constant term -2 follows a simpler rule, where the integral of a constant times \( t \) is the constant times the integral of \( t \) which is \( t \) itself.
The general form of the power rule for indefinite integrals is \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \) for all values of \( n \) not equal to -1. Here, \( C \) represents the constant of integration, which is added because indefinite integrals represent a family of functions.
In our exercise, when we apply the power rule to the term \( t^3 \), we raise the exponent by 1, resulting in \( t^4 \) and divide by the new exponent, yielding \( \frac{t^4}{4} \) as part of our indefinite integral. The integration of the constant term -2 follows a simpler rule, where the integral of a constant times \( t \) is the constant times the integral of \( t \) which is \( t \) itself.
Integral Simplification
Before we can apply integration techniques, it is often necessary to simplify the function we want to integrate. This process can involve expanding products, simplifying fractions, or trigonometric identities, among others, to bring the integrand into a form amenable to standard integration methods.
For the problem at hand, simplification involves expanding the binomial \( t^2(t - \frac{2}{t}) \) to get \( t^3 - 2 \). This expansion turns a product of terms into a sum of terms, making it straightforward to apply the power rule as described in the previous section.
Simplification is a crucial step; without it, we might not be able to integrate the function directly or might end up with a more complex integral than necessary. In our example, by simplifying the integrand first, we could easily recognize the terms to which we could apply the power rule individually.
For the problem at hand, simplification involves expanding the binomial \( t^2(t - \frac{2}{t}) \) to get \( t^3 - 2 \). This expansion turns a product of terms into a sum of terms, making it straightforward to apply the power rule as described in the previous section.
Simplification is a crucial step; without it, we might not be able to integrate the function directly or might end up with a more complex integral than necessary. In our example, by simplifying the integrand first, we could easily recognize the terms to which we could apply the power rule individually.
Verification by Differentiation
After finding the indefinite integral, it's a good practice to verify the accuracy of our result by differentiating it and checking if we obtain the original function. This step acts as a confirmation that we integrated correctly.
To verify the result \( \frac{t^4}{4} - 2t + C \) from our exercise, we differentiate each term to get \( t^3 \) from \( \frac{t^4}{4} \) and -2 from \( -2t \). The constant of integration \( C \) vanishes upon differentiation as the derivative of any constant is zero.
When we piece together the derivatives of the individual terms, we're left with the function \( t^3 - 2 \) which matches our original integrand. If our differentiation did not match the original integrand, it would indicate an error in our integration process, prompting us to revisit our steps.
To verify the result \( \frac{t^4}{4} - 2t + C \) from our exercise, we differentiate each term to get \( t^3 \) from \( \frac{t^4}{4} \) and -2 from \( -2t \). The constant of integration \( C \) vanishes upon differentiation as the derivative of any constant is zero.
When we piece together the derivatives of the individual terms, we're left with the function \( t^3 - 2 \) which matches our original integrand. If our differentiation did not match the original integrand, it would indicate an error in our integration process, prompting us to revisit our steps.
Other exercises in this chapter
Problem 21
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Evaluate the definite integral of the transcendental function. Use a graphing utility to verify your result. $$ \int_{-\pi / 6}^{\pi / 6} \sec ^{2} x d x $$
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Find the indefinite integral. $$ \int \frac{\cos t}{1+\sin t} d t $$
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Find the derivative of the function. \(h(t)=t-\operatorname{coth} t\)
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