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Chapter 4: Problem 5

Evaluate. Assume \(u>0\) when In \(u\) appears. (Be sure to check by differentiating!) $$\int\left(3 t^{4}+2\right) t^{3} d t$$

Short Answer

Expert verified
\( \frac{3t^8}{8} + \frac{t^4}{2} + C \)

Step by step solution

01

Recognize the Integral Form

Identify the given integral: \[\begin{equation}\int (3t^4 + 2)t^3 \, dt \end{equation}\]Notice the integral is in the form of a product of a polynomial function and its derivative.
02

Simplify the Integrand

Rewrite the integrand if necessary. Here, distribute the polynomial: \[\begin{equation}(3t^4 + 2)t^3 = 3t^4 \times t^3 + 2 \times t^3 = 3t^7 + 2t^3 \end{equation}\]Thus, the integral becomes: \[\begin{equation}\int (3t^7 + 2t^3) \, dt \end{equation}\]
03

Integrate Each Term

Integrate each term separately: For \[\begin{equation}3t^7 \end{equation}\]: \[\begin{equation}\int 3t^7 \, dt = 3 \frac{t^8}{8} = \frac{3t^8}{8} \end{equation}\] For \[\begin{equation}2t^3 \end{equation}\]: \[\begin{equation}\int 2t^3 \, dt = 2 \frac{t^4}{4} = \frac{2t^4}{4} = \frac{t^4}{2} \end{equation}\]
04

Combine the Results

Combine the integrals to obtain the final result: \[\begin{equation}\frac{3t^8}{8} + \frac{t^4}{2} + C \end{equation}\]
05

Verify by Differentiating

Differentiate the result to check if it yields the original integrand: \[\begin{equation}\frac{d}{dt} \left( \frac{3t^8}{8} + \frac{t^4}{2} + C \right) = \frac{3 \times 8t^7}{8} + \frac{4t^3}{2} = 3t^7 + 2t^3 \end{equation}\] The differentiated form matches the original integrand, confirming the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Integration
Polynomial integration involves finding the integral of polynomial expressions. A polynomial is a mathematical expression consisting of variables and coefficients, arranged in terms of increasing or decreasing powers of the variable. In our exercise, we deal with the polynomial \( 3t^4 + 2 \) multiplied by \( t^3 \).

To integrate polynomials, follow these steps:
  • Distribute terms if necessary, so each term is in the form \( at^n \).
  • Use the power rule of integration: \(\frac{t^{n+1}}{n+1}\text{ where } n eq -1\).
  • Combine the integrals of each term.
In our example:
  • First, rewrite the integrand: \( (3t^4 + 2)t^3 \), by distributing it to get \( 3t^7 + 2t^3 \).
  • Then, integrate each term separately: \(\frac{3t^8}{8}\text{ and }\frac{t^4}{2}\).
  • The integrated result is combined: \( \frac{3t^8}{8} + \frac{t^4}{2} + C \).
Definite and Indefinite Integrals
An integral can be definite or indefinite. An indefinite integral, which is what we solved in our exercise, represents a family of functions and includes an arbitrary constant, indicated by \( + C \).

Indefinite integrals are used to find antiderivatives. Here's how to write them:
  • The general format is \( \int f(x) \, dx = F(x) + C \), where \( F(x) \) is the antiderivative.
  • The constant \( C \) represents any constant because differentiation of a constant is zero.
Definite integrals, however, involve specific bounds of integration. They calculate the area under the curve from one point to another, resulting in a specific number. The format for definite integrals is:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a). \]

In our exercise, we only deal with the indefinite integral of \( (3t^4 + 2)t^3 \), leading to \( \frac{3t^8}{8} + \frac{t^4}{2} + C \).
Differentiation
Differentiation is finding the derivative of a function. It's the reverse process of integration. We differentiate to check your work as done in the given exercise. Differentiation gives the rate of change of a function concerning its variable.

To differentiate the integrated result \( \frac{3t^8}{8} + \frac{t^4}{2} + C \):
  • Use the power rule for differentiation: \(\frac{d}{dt}(t^n) = nt^{n-1}\).
Let's apply this step-by-step:
  • For \( \frac{3t^8}{8} \): Multiply by the exponent and reduce it by one: \( \frac{3 \times 8t^7}{8} = 3t^7 \).
  • For \( \frac{t^4}{2} \): Similarly, \( \frac{4 \times t^3}{2} = 2t^3 \).
  • The constant \( C \) vanishes, as it's derivative is zero.
Combining these results, we get: \( 3t^7 + 2t^3 \), which matches the original integrand. Therefore, our integration is verified accurately.

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