Problem 6 - Calculus of a Single Variable

Question

Choosing a Formula In Exercises \(5-14\) , select the basic integration formula you can use to find the integral, and identify \(u\) and \(a\) when appropriate. $$ \int \frac{2 t+1}{t^{2}+t-4} d t $$

Step-by-Step Solution

Answer

The integral \(\int \frac{2t+1}{t^{2}+t-4} dt\) can be solved using the formula \(\int \frac{f'}{f} dt=\ln |f| + C\), with \(u = t^2 + t - 4\) and \(a = 1\), and results in \(\ln |t^2 + t - 4| + C\).

Step 1: Identify the Formula

The integral has the form \(\int \frac{f'}{f} dt\), where \(f' = 2t+1\) and \(f = t^{2}+t-4\). The integral can therefore be solved using the formula \(\int \frac{f'}{f} dt = \ln |f| + C\) .

Step 2: Identify u and a

Here, the variable \(u\) can be identified as \(u=t^2 +t - 4\) as it is the function at the denominator, and \(a = 1\) as there is no scaling factor for \(u\) in the integral.

Step 3: Apply the Formula

Applying the formula from step 1 using the identifications from step 2 results in the following integral: \(\int \frac{2t+1}{t^2 + t - 4} dt = \ln |t^2 + t - 4| + C\).

Key Concepts

Basic Integration Formula

The basic integration formula is a cornerstone of calculus. It helps us to solve integrals of functions that fit certain standard forms.

For example, the integral \[\int \frac{f'}{f} \, dt \]can be solved using the basic logarithmic integration, yielding:\[\ln |f| + C\]where \(C\) is the constant of integration. This is particularly useful when the integrand is the derivative of its denominator.

In the given exercise, we have:

• Numerator \(f' = 2t+1\)
• Denominator \(f = t^{2}+t-4\)

Recognizing this form guides the application of the formula directly to solve the integral.

Integration by Substitution

Integration by substitution is akin to the reverse of differentiating using the chain rule.

This method simplifies integration by replacing a variable with another to make the problem easier. In many cases, this involves identifying a function and its derivative in the integral, which helps in substituting variables effectively.

• In our exercise, identifying \(f = t^2 + t - 4\) helps us see the structure of the integral.
• This allows us to set \(u = t^2 + t - 4\).
• Since \(f' = 2t+1\) matches the numerator, substitution can proceed seamlessly.

The integration by substitution technique transforms a complex integral into a more manageable one by changing the variable.

Logarithmic Integration

Logarithmic integration is a specific technique used when integrating rational functions, particularly when the form fits \(\frac{f'}{f}\).

This method results in expressions that are natural logarithms due to the derivative-numerator structure. In simpler terms, if the derivative of the denominator is present in the numerator, the integral can be easily expressed using logarithms:

• Here, \(\int \frac{2t+1}{t^{2}+t-4} dt\) fits perfectly into this form.
• We use the basic formula \(\ln |f| + C\).

Thus, this approach turns challenging integrals into familiar logarithmic forms, simplifying the solution process and helping in handling complex algebraic expressions within an integral.