Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate the indicated indefinite integrals. $$ \int\left(t^{2}-2 \cos t\right) d t $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral

The question asks us to evaluate the indefinite integral \(\int (t^2 - 2\cos t) \, dt\). This is a linear combination of two simpler functions: \(t^2\) and \(-2\cos t\).

Step 2: Apply the Sum Rule for Integration

The integral of a sum of functions equals the sum of their integrals. Thus, we can write:\[\int (t^2 - 2 \cos t) \, dt = \int t^2 \, dt - 2 \int \cos t \, dt\]

Step 3: Integrate \(t^2\)

To integrate \(t^2\), use the power rule: \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\), so\[\int t^2 \, dt = \frac{t^3}{3} + C_1\]

Step 4: Integrate \(-2 \cos t\)

The integral of \(\cos t\) is \(\sin t\). Thus, \[2 \int \cos t \, dt = 2 \sin t + C_2\]

Step 5: Combine the Results

Combine the results of Step 3 and Step 4. Remember that the unknown constants \(C_1\) and \(C_2\) can be combined into a single arbitrary constant \(C\).\[\int (t^2 - 2 \cos t) \, dt = \frac{t^3}{3} - 2 \sin t + C\]

Step 6: Write the Final Answer

The evaluated indefinite integral is: \[\frac{t^3}{3} - 2 \sin t + C\] where \(C\) is the integration constant.

Key Concepts

Integration Techniques

Integration is a method used to find the accumulated area under a curve, simplifying complex mathematical expressions. When dealing with an integral like \[ \int (t^2 - 2 \, \cos t) \, dt \]we need to identify which integration techniques might help us. In this problem, we're dealing with a combination of polynomial and trigonometric functions. The trick is to simplify the integral by breaking it down into easily manageable parts.
To do this:

• Identify the types of functions present. In this case, polynomial and trigonometric.
• Use the sum rule, which allows you to integrate each part separately.
• Choose a specific integration technique based on the type of function, like the power rule for polynomials and basic trigonometric integration for trigonometric functions.

By breaking down the integral in this way, it's easier to manage and solve each part individually.

Power Rule

The power rule is a fundamental technique for integrating polynomial functions. It states that to integrate a power of a variable, you increase its exponent by 1 and divide by the new exponent. This rule is applied as follows:\[\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\]For instance, in our problem,\[\int t^2 \, dt = \frac{t^3}{3} + C_1\]It's crucial to remember:

• This rule does not apply when \( n = -1 \).
• Always add the constant of integration \( C \) at the end of an indefinite integral.

The power rule is straightforward but immensely useful, providing a quick means to integrate terms that are powers of the variable.

Trigonometric Integration

Trigonometric integration deals with integrating trigonometric functions like sine, cosine, and tangent. In this exercise, we have the term \( -2 \cos t \).

To integrate this, recall the basic formula:\[\int \cos t \, dt = \sin t + C_2\]When combined with a constant multiplier, like \(-2\),we have:\[-2 \int \cos t \, dt = -2 (\sin t) = -2 \sin t + C_2\]A couple of essential tips:

• Always keep track of coefficients when integrating.
• Remember to include the constant of integration \( C \) at the end of the process.

Understanding equations of this type is crucial because trigonometric functions often appear in various areas of mathematics and physics.

Constant of Integration

Whenever you calculate an indefinite integral, always add the arbitrary constant of integration, denoted as \( C \). This constant represents an infinite family of functions that differ by a constant. In our revised expression:\[\frac{t^3}{3} - 2 \sin t + C\]\( C \) is critical because:

• The derivative of any constant is zero, so it "disappears" when taking derivatives.
• The constant ensures that the integral remains general until a specific condition or initial value is given.

It reflects the idea that there are many potential solutions to an indefinite integral problem, each differing by this constant. Ensuring that \( C \) is always included makes your solutions comprehensive and accurate.