Problem 39 - Applied Calculus

Question

Find the indefinite integrals. $$ \int\left(t^{2}+5 t+1\right) d t $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{t^3}{3} + \frac{5t^2}{2} + t + C \).

Step 1: Split the Integral

To solve the integral \( \int (t^2 + 5t + 1) \, dt \), start by splitting it into separate integrals: \( \int t^2 \, dt + \int 5t \, dt + \int 1 \, dt \). This simplification allows us to integrate each term individually.

Step 2: Integrate the First Term

Find the integral of the first term \( \int t^2 \, dt \). The antiderivative of \( t^2 \) is \( \frac{t^3}{3} \). This follows from the power rule for integration \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \).

Step 3: Integrate the Second Term

Next, integrate the second term \( \int 5t \, dt \). The antiderivative of \( 5t \) is \( \frac{5t^2}{2} \). You factor out the 5 and apply the power rule \( \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \) with \( n=1 \).

Step 4: Integrate the Third Term

Now, integrate the constant term \( \int 1 \, dt \). The antiderivative of 1 is \( t \). This is because the derivative of \( t \) is 1.

Step 5: Combine the Results

Finally, combine the results of each integral. The indefinite integral is \( \frac{t^3}{3} + \frac{5t^2}{2} + t + C \), where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

The Power Rule for Integration is a simple yet powerful tool that helps us find antiderivatives of polynomial functions easily. It states that the integral of a power of a variable, such as \( t^n \), can be found using the formula:\[\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\]where \( n eq -1 \) and \( C \) is the constant of integration. This rule is derived from reversing the process of differentiation of power functions, making integration straightforward.

• Application: To apply this rule, simply increase the exponent by one and divide by the new exponent. Don’t forget to add \( C \).
• Example: For \( t^2 \), the integral is \( \frac{t^3}{3} + C \), since the new exponent is 3.

Using the Power Rule simplifies problems with derivatives, especially those involving polynomials, by allowing us to integrate term by term.

Antiderivatives

Antiderivatives, also known as primitives, are functions whose derivatives give the original function. For any given function \( f(t) \), an antiderivative is a function \( F(t) \) such that \( F'(t) = f(t) \). The act of finding antiderivatives is referred to as integration.

• Understanding: Think of antiderivatives as the reverse process of differentiation. Instead of finding the slope or rate of change, you're discovering the function before it was differentiated.
• General Form: Every nontrivial antiderivative includes a constant of integration \( C \), representing any potential vertical shift of the original function.

For instance, when integrating \( t^2 \), you find \( \frac{t^3}{3} + C \) because the derivative of \( \frac{t^3}{3} \) brings you back to \( t^2 \). Antiderivatives are essential in determining areas under curves and in solving differential equations.

Integration of Polynomials

Integrating polynomials involves applying the Power Rule to each term of the polynomial separately. Polynomials are expressions composed of variables raised to a power and characterized by their coefficients, such as \( a_n t^n + a_{n-1} t^{n-1} + \ldots + a_0 \).

• Step-by-Step: Break down the polynomial into individual terms and integrate each one individually. Use the Power Rule for each term and remember to factor out constants.
• Example: For the integral \( \int (t^2 + 5t + 1) \, dt \):
  • Integrate \( t^2 \) to get \( \frac{t^3}{3} \)
  • Integrate \( 5t \) to achieve \( \frac{5t^2}{2} \)
  • Integrate \( 1 \) to obtain \( t \)
• Result: Combine the results: \( \frac{t^3}{3} + \frac{5t^2}{2} + t + C \), where \( C \) is the constant of integration.

This approach makes the integration process manageable, enhancing accuracy and understanding, making it particularly useful in calculus and further mathematical applications.