Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \int\left(t^{2}-2 \cos t\right) d t $$

Step-by-Step Solution

Answer

\( \frac{t^3}{3} - 2 \sin t + C \)

Step 1: Break Down the Integral

The given integral is \( \int (t^2 - 2 \cos t) \, dt \). We can split this into two separate integrals: \( \int t^2 \, dt \) and \( -2 \int \cos t \, dt \). This allows us to tackle each part individually.

Step 2: Integrate \(t^2\)

The integral of \( t^2 \) with respect to \( t \) is \( \frac{t^3}{3} \). This follows from the power rule of integration, \( \int t^n \, dt = \frac{t^{n+1}}{n+1} \), where \( n = 2 \).

Step 3: Integrate \(-2\cos t\)

The integral of \(-2 \cos t\) is \(-2 \sin t\). This is because the integral of \( \cos t \) is \( \sin t \). Thus, \(-2 \int \cos t \ dt = -2 \sin t\).

Step 4: Combine Results

Now, combine the results of the integrals: \( \frac{t^3}{3} - 2 \sin t + C \), where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of functions. These techniques can transform complex problems into simpler ones, making it easier to solve them. In the exercise given above, one effective integration technique applied is to split the integral into smaller parts.
This strategy is known as the **Sum Rule for Integration**. The Sum Rule states that the integral of a difference or sum of functions can be broken down into the sum or difference of the integrals of each function individually.
By applying this technique, we are able to decompose the given function \((t^2 - 2 \cos t)\) into two separate parts: \(t^2\) and \(-2 \cos t\). Each term is then integrated separately. Breaking it down makes the process of finding the integrals significantly easier.
This technique is particularly useful when dealing with polynomials and trigonometric functions in combination, allowing for straightforward calculation by addressing each component on its own.

Power Rule

The power rule for integration simplifies finding the antiderivative of polynomial expressions. It's one of the most fundamental integration techniques you will encounter. The rule states that for any function \(t^n\), its integral is given by \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\), where \(n eq -1\).
In our example, we have the function \(t^2\) as part of the integrand. Applying the power rule helps us find its integral efficiently. Here, \(n = 2\), so according to the power rule, the integral of \(t^2\) is:\[\frac{t^3}{3} + C\]
This makes resolving powers of \(t\) into integrals straightforward, saving time and effort during calculations. The application of this rule is intuitive after some practice, and it is highly useful in solving integration problems involving polynomials.

Trigonometric Integrals

Trigonometric integrals involve functions like sin, cos, and tan, among others. These integrals often arise in calculus problems and require specific techniques for their calculation. In the given exercise, we deal with \(-2 \cos t\).
For trigonometric functions, it's crucial to remember their integrals. The basic integral of \(\cos t\) is \(\sin t\), and thus \(-2 \cos t\) integrates to \(-2 \sin t\). The negative coefficient remains outside the integral, making the calculation straightforward:

• Integral of \(\cos t\) is \(\sin t\)
• Coefficient of \(-2\) means multiplying the integral by \(-2\)
• Result: \(-2 \int \cos t \, dt = -2 \sin t\)

This particular understanding of trigonometric integrals can greatly ease various calculus computations, focusing on recognizing patterns and applying known integrals to solve exercises effectively.