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Magazine Chapter 16: Problem 25
Evaluate each integral. $$\int t^{2} \sqrt{2 t^{3}-7} d t$$
Short Answer
Expert verified
The integral evaluates to \( \frac{1}{9} (2t^3 - 7)^{3/2} + C \).
Step by step solution
01
Choose a Substitution
To evaluate the integral \( \int t^{2} \sqrt{2 t^{3} - 7} \, d t \), we look for a substitution that simplifies the expression inside the integral. Let \( u = 2t^3 - 7 \). The derivative of \( u \) is \( du = 6t^2 \, dt \). Rearranging gives us \( t^2 \, dt = \frac{1}{6} \, du \).
02
Substitute and Simplify
Using our substitution \( u = 2t^3 - 7 \), the integral becomes \( \int t^2 \sqrt{2t^3 - 7} \, dt = \int \sqrt{u} \cdot \frac{1}{6} \, du \). Simplify this to \( \frac{1}{6} \int u^{1/2} \, du \).
03
Integrate with Respect to u
Now we integrate \( \int u^{1/2} \, du \). The antiderivative of \( u^{1/2} \) is \( \frac{2}{3} u^{3/2} \). Therefore, \( \frac{1}{6} \int u^{1/2} \, du = \frac{1}{6} \cdot \frac{2}{3} u^{3/2} = \frac{1}{9} u^{3/2} \).
04
Substitute Back in Terms of t
Replace \( u \) with the original expression in terms of \( t \). We have \( u = 2t^3 - 7 \). Thus, \( \frac{1}{9} u^{3/2} = \frac{1}{9} (2t^3 - 7)^{3/2} \).
05
Add the Constant of Integration
Finally, add the constant of integration \( C \) to our result. The solution to the integral is \( \frac{1}{9} (2t^3 - 7)^{3/2} + C \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Substitution Method
The substitution method is a handy technique in integral calculus that helps to simplify complex integrals. In our exercise, solving the integral \( \int t^{2} \sqrt{2 t^{3}-7} \, dt \) requires us to use substitution.
By transforming the integral into a more manageable form, we can find the antiderivative more easily. Here's how it works:
By transforming the integral into a more manageable form, we can find the antiderivative more easily. Here's how it works:
- Choose a Substitution: The tricky part is identifying what to substitute for. We look for expressions inside the integral that, if replaced, can reduce its complexity. In this case, we choose \( u = 2t^3 - 7 \). This choice is wise because the derivative of \( u \), which is \( du = 6t^2 \, dt \), relates directly to parts of the original integrand.
- Rewrite the Integral: Using our substitution, you replace \( t^2 \, dt \) with \( \frac{1}{6} du \). This transforms the integral from a function of \( t \) into a function of \( u \): \( \frac{1}{6} \int u^{1/2} \, du \). The substitution method thus simplifies the original problem considerably.
Antiderivative
In calculus, finding an antiderivative, also known as an indefinite integral, is like running integration backwards to discover a function whose derivative gives us the original integrand.
In our task, we simplified the integral to \( \frac{1}{6} \int u^{1/2} \, du \). To find its antiderivative, we use the formula for integrating power functions:
In our task, we simplified the integral to \( \frac{1}{6} \int u^{1/2} \, du \). To find its antiderivative, we use the formula for integrating power functions:
- Power Rule for Integration: If \( u^n \) is our integrand, the antiderivative is \( \frac{u^{n+1}}{n+1} \) plus a constant \( C \), assuming \( n eq -1 \). For \( u^{1/2} \), this becomes \( \frac{2}{3} u^{3/2} \).
- Combining with Coefficients: Multiply by the coefficient previously factored out, \( \frac{1}{6} \), leading to \( \frac{1}{9} u^{3/2} \).
Constant of Integration
The constant of integration is a crucial element in indefinite integrals and represents the family of all antiderivatives of a function.
After integrating \( \frac{1}{6} \int u^{1/2} \, du \) and simplifying, our result was \( \frac{1}{9} (2t^3 - 7)^{3/2} \). The next important step involves adding the constant of integration, \( C \):
After integrating \( \frac{1}{6} \int u^{1/2} \, du \) and simplifying, our result was \( \frac{1}{9} (2t^3 - 7)^{3/2} \). The next important step involves adding the constant of integration, \( C \):
- Why Add \( C \)? When antiderivatives are found, any constant could be added to them and still result in the same derivative. Thus, the constant accounts for all vertical shifts of the antiderivative graph.
- Computed Solution: Our final expression becomes \( \frac{1}{9} (2t^3 - 7)^{3/2} + C \), ensuring our solution encompasses all potential forms of the antiderivative.
