Problem 116 - Calculus for Biology and Medicine

Question

Evaluate the definite integrals. $$ \int_{0}^{2} 2 t e^{t^{2}} d t $$

Step-by-Step Solution

Answer

The integral evaluates to \( e^4 - 1 \).

Step 1: Identify the function to integrate

The given integral is \( \int_{0}^{2} 2t e^{t^2} \, dt \). Notice that it involves an exponential function \( e^{t^2} \) and a polynomial \( 2t \).

Step 2: Choose a suitable integration method

Given that the derivative of \( t^2 \) is \( 2t \), this suggests a substitution method might be effective. We will use the substitution \( u = t^2 \).

Step 3: Perform the substitution

Set \( u = t^2 \), so that \( du = 2t \, dt \). Thus, \( 2t \, dt = du \). The limits of integration will change according to \( u = t^2 \): when \( t = 0 \), \( u = 0 \); and when \( t = 2 \), \( u = 4 \).

Step 4: Rewrite the integral in terms of \( u \)

Substitute into the original integral to get the new integral in terms of \( u \):\[ \int_{0}^{4} e^{u} \, du \]

Step 5: Integrate with respect to \( u \)

The integral of \( e^{u} \) with respect to \( u \) is \( e^{u} \). Therefore, the integral becomes:\[ \left[ e^{u} \right]_{0}^{4} \]

Step 6: Evaluate the definite integral

Apply the limits of integration:\[ e^{4} - e^{0} = e^4 - 1 \]

Step 7: Conclusion

The value of the definite integral \( \int_{0}^{2} 2t e^{t^2} \, dt \) is \( e^4 - 1 \).

Key Concepts

Substitution Method

The substitution method is a powerful technique used in integration to simplify integrals. It is especially useful when dealing with integrals that contain compositions of functions, such as a polynomial multiplied by an exponential function. In this case, the function inside the exponential is a square function, and its derivative appears elsewhere in the integral. With substitution, we aim to make the integral neater by transforming it into a standard form that is easier to solve.

Here's how it works:

• Choose a substitution: Pick a function within the integral to substitute with a new variable. For this exercise, we used the substitution \( u = t^2 \).
• Find the differential: Derive the substitution to find \( du \). From \( u = t^2 \), we get \( du = 2t \, dt \).
• Change the limits: Substitute the new variable into the original limits to find the new integration limits. For \( t = 0 \), \( u = 0 \); and for \( t = 2 \), \( u = 4 \).

This substitution changes the integral into a simpler form, allowing us to solve the integral with respect to one simple variable, \( u \). By converting complex expressions into simpler ones, substitution helps streamline the integration process.

Exponential Function

Exponential functions are a fundamental concept in mathematics, frequently encountered in calculus. They have the form \( e^{x} \), where \( e \) is the base of natural logarithms, approximately equal to 2.71828. Exponential functions are unique because they have the same derivative and integral form, which simplifies calculations involving them.

In our exercise, the function \( e^{t^2} \) appears inside the integrand. With the substitution \( u = t^2 \), this part of the integral is transformed to \( e^{u} \). The characteristic property of exponential functions is leveraged here to easily compute the integral. The integral of an exponential function \( e^{u} \) is straightforward, with its integral being simply:

• \( \int e^{u} \, du = e^{u} + C \)

For a definite integral, the constant of integration \( C \) is ignored, allowing us to apply the limits directly after integration. This intrinsic property of exponential functions greatly simplifies their integration.

Integration Limits

Integration limits are critical when evaluating definite integrals, as they define the interval over which the function is being integrated. They are denoted by the values at the top and bottom of the integral symbol, \( \int_{a}^{b} \). In substitution, we must remember to change these limits according to the chosen substitution.

For the original integral \( \int_{0}^{2} 2t e^{t^2} \, dt \), the limits of integration are from 0 to 2 for \( t \). After substituting \( u = t^2 \), these limits need updating to reflect the new variable \( u \).

For example:

• When \( t = 0 \), \( u = 0^2 = 0 \).
• When \( t = 2 \), \( u = 2^2 = 4 \).

Thus, the new limits for \( u \) are from 0 to 4. This update is crucial for accurately evaluating the definite integral once the integration process is complete. Always ensure that the integration limits align with the variable being integrated, which guarantees the correct calculation of the area under the curve within the specified bounds.