Problem 39 - Biocalculus Calculus for the Life Sciences

Question

Find the general indefinite integral. \(\int(1-t)\left(2+t^{2}\right) d t\)

Step-by-Step Solution

Answer

\( \int (1-t)(2+t^2) \, dt = 2t + \frac{t^3}{3} - t^2 - \frac{t^4}{4} + C \).

Step 1: Expand the Integrand

First, we need to expand the expression \( (1-t)(2+t^2) \) using distribution. This gives us \( 2 + t^2 - 2t - t^3 \).

Step 2: Write the Integral with Expanded Terms

Re-write the integral with the expanded terms: \( \int (2 + t^2 - 2t - t^3) \, dt \).

Step 3: Integrate Each Term Separately

Integrate each term separately: \(\int 2 \, dt = 2t + C_1\), \(\int t^2 \, dt = \frac{t^3}{3} + C_2\), \(\int -2t \, dt = -t^2 + C_3\), \(\int -t^3 \, dt = -\frac{t^4}{4} + C_4\).

Step 4: Combine the Integrals

Add the results of integrating each term together, incorporating the constants of integration:\[ 2t + \frac{t^3}{3} - t^2 - \frac{t^4}{4} + C \] where \( C = C_1 + C_2 + C_3 + C_4 \) is the general constant of integration.

Key Concepts

Expansion of Expressions

The expansion of expressions is a valuable step in solving mathematical problems, particularly in calculus where multiple components of an expression need to be integrated or differentiated. In our example, the expression \((1-t)(2+t^2)\) is expanded using distribution. This process involves multiplying each term in the first parenthesis by each term in the second. Here's how it's done:

• Multiply 1 by 2, resulting in 2.
• Multiply 1 by \(t^2\), resulting in \(t^2\).
• Multiply \(-t\) by 2, resulting in \(-2t\).
• Multiply \(-t\) by \(t^2\), resulting in \(-t^3\).

Combining all these results, the expanded form of the expression becomes \(2 + t^2 - 2t - t^3\). This is now in a friendly format that makes integration much easier.

Integration Techniques

Integrating functions involves finding the function whose derivative is the given function. There are several techniques for integration, depending on the form of the integrand. For simple polynomials like \(2 + t^2 - 2t - t^3\), we use basic integration rules which correspond to reversing the process of differentiation. Each term in the polynomial is integrated separately, and the integral of \(t^n\)is \(\frac{t^{n+1}}{n+1} + C\).Let's integrate each term in \(2 + t^2 - 2t - t^3\):

• \(\int 2 \, dt\) becomes \(2t + C_1\).
• \(\int t^2 \, dt\) becomes \(\frac{t^3}{3} + C_2\).
• \(\int (-2t) \, dt\) becomes \(-t^2 + C_3\).
• \(\int (-t^3) \, dt\) becomes \(-\frac{t^4}{4} + C_4\).

These "antiderivatives" are then combined to form the integrated expression.

Constant of Integration

The constant of integration is a crucial component of indefinite integrals. When integrating a function, the result is a family of functions, all differing by a constant. This is because differentiation of a constant yields zero, and hence it is impossible to determine this constant when integrating.To express this, we sum the individual constants from each integrated term into a single constant, \(C = C_1 + C_2 + C_3 + C_4\). This represents all possible vertical shifts of the antiderivative's graph. In our final integrated result, \(2t + \frac{t^3}{3} - t^2 - \frac{t^4}{4} + C\), \(C\) gives us flexibility when matching boundary conditions or specific solutions. It highlights the unlimited number of solutions that fit the original differential equation.