Problem 10 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the indefinite integral. \(\int(3 t+2)^{2.4} d t\)

Step-by-Step Solution

Answer

The indefinite integral is \(\frac{(3t+2)^{3.4}}{10.2} + C\).

Step 1: Choose a Substitution

Notice that the integrand in the form \((3t+2)^{2.4} \) can be simplified with a substitution. Let \( u = 3t + 2\). This means \( du = 3 \, dt\) or \( dt = \frac{1}{3} \, du\). Perform this substitution in the integral.

Step 2: Substitute and Simplify the Integral

Substitute \( u = 3t + 2\) into the integral and replace \( dt \) with \( \frac{1}{3} \, du\). The integral becomes: \[ \int (3t+2)^{2.4} \, dt = \int u^{2.4} \cdot \frac{1}{3} \, du = \frac{1}{3} \int u^{2.4} \, du \]

Step 3: Integrate with Respect to u

Integrate \( u^{2.4} \) with respect to \( u \): \[ \int u^{2.4} \, du = \frac{u^{2.4+1}}{2.4+1} = \frac{u^{3.4}}{3.4} \]Multiplying by \( \frac{1}{3} \) from Step 2, we get:\[ \frac{1}{3} \cdot \frac{u^{3.4}}{3.4} = \frac{u^{3.4}}{10.2} \]

Step 4: Substitute Back in Terms of t

Replace \( u \) with the original variable to obtain the antiderivative in terms of \( t \): \[ \frac{(3t+2)^{3.4}}{10.2} \]

Step 5: Add the Constant of Integration

Finally, add the constant of integration \( C \) to complete the indefinite integral solution: \[ \int (3t+2)^{2.4} \, dt = \frac{(3t+2)^{3.4}}{10.2} + C \]

Key Concepts

Substitution Method

When dealing with complex integrals such as \( \int(3t+2)^{2.4} dt \), the substitution method can simplify the process. This technique involves finding a substitution that transforms a difficult integral into an easier one.
In our example, we let \( u = 3t + 2 \). By doing this, the derivative \( du/dt \) is 3, leading to \( du = 3 \, dt \), or equivalently, \( dt = \frac{1}{3} \, du \).
This step is crucial as it rewrites the integral in terms of \( u \), simplifying the expression. We substitute \( u \) for \( 3t + 2 \) and replace \( dt \) to get \( \int u^{2.4} \cdot \frac{1}{3} \, du \). This transformation makes the integral easier to manage.

• The power of the substitution method lies in its ability to simplify integrands.
• Helps in matching the integral to a basic formula or a known result.

For students, mastering this technique enhances their ability to tackle a variety of integration problems with confidence.

Integration Techniques

Integration is a key part of calculus and comes with several techniques, each suitable for different scenarios. The substitution method is just one of many.
Understanding these integration techniques is essential:

• Basic Integration: The process of finding antiderivatives or integrals of routine functions like polynomials.
• Substitution Method: Useful when the integrand can be simplified by changing variables.
• Integration by Parts: Applies to integrals of products, derived from the product rule of differentiation.

In our scenario, after substituting \( u = 3t + 2 \), the integral \( \frac{1}{3} \int u^{2.4} \, du \) is straightforward since it mimics the basic power rule for integration:
\[ \int u^{n} \, du = \frac{u^{n+1}}{n+1} \]
Thus, we apply this rule to \( u^{2.4} \) yielding \( \frac{u^{3.4}}{3.4} \). Knowing when and how to apply each technique can greatly aid in solving integrals encountered in various disciplines.

Calculus for Life Sciences

Calculus offers powerful tools for modeling and solving problems in life sciences. Integration, for instance, helps calculate quantities like population growth, medication dosage over time, and area under curves representing biological phenomena.
In life sciences, calculus assists in:

• Understanding rates of change such as how rapidly a tumor grows or how fast a drug disperses in the bloodstream.
• Modeling population dynamics through differential equations.
• Analyzing biochemical reactions over time.

The integration encountered in the equation \( \int (3t+2)^{2.4} dt \) could, for example, represent a simplified model where \( t \) is time and \( (3t+2)^{2.4} \) is related to a biological process affecting growth or decay rate.
Life sciences students leverage these mathematical strategies to make informed decisions and predictions in experiments and research, illustrating the cross-disciplinary importance of calculus.