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Magazine Chapter 4: Problem 28
Evaluate the indicated indefinite integrals. $$ \int\left(t^{2}-2 \cos t\right) d t $$
Short Answer
Expert verified
\(\frac{t^3}{3} - 2 \sin t + C\)
Step by step solution
01
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 \(-2\cos t\).
02
Apply the Sum Rule for Integration
The integral of a sum of functions equals the sum of their integrals. Thus, we can write:\[\int (t^2 - 2 \cos t) \, dt = \int t^2 \, dt - 2 \int \cos t \, dt\]
03
Integrate \(t^2\)
To integrate \(t^2\), use the power rule: \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\), so\[\int t^2 \, dt = \frac{t^3}{3} + C_1\]
04
Integrate \(-2 \cos t\)
The integral of \(\cos t\) is \(\sin t\). Thus, \[2 \int \cos t \, dt = 2 \sin t + C_2\]
05
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 \(C\).\[\int (t^2 - 2 \cos t) \, dt = \frac{t^3}{3} - 2 \sin t + C\]
06
Write the Final Answer
The evaluated indefinite integral is: \[\frac{t^3}{3} - 2 \sin t + C\] where \(C\) is the integration constant.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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.
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:
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.
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.
