Problem 48 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(e^{2 t}+2 \sqrt{t}\right) d t$$

Step-by-Step Solution

Answer

The indefinite integral of the function \(e^{2 t}+2 \sqrt{t}\) is \(\frac{1}{2}e^{2t} + \frac{4}{3}t^{3/2} + C\), where C is an arbitrary constant.

Step 1: Integrate the first term \(e^{2t}\)

To integrate \(e^{2t}\), we can use the formula \(\int e^{at} dt = \frac{1}{a}e^{at} + C\). In this case, \(a = 2\): $$\int e^{2t} dt = \frac{1}{2}e^{2t} + C_1$$

Step 2: Integrate the second term \(2\sqrt{t}\)

To integrate \(2\sqrt{t}\), we can first rewrite the function as \(2t^{\frac{1}{2}}\). Then, we can use the power rule for integration, \(\int t^n dt = \frac{1}{n+1}t^{n+1} + C\): $$\int 2t^{\frac{1}{2}} dt = 2\int t^{\frac{1}{2}} dt = 2\left(\frac{1}{\frac{3}{2}}t^{\frac{3}{2}} + C_2\right) = \frac{4}{3}t^{\frac{3}{2}} + C_2$$

Step 3: Combine the results

Combining the antiderivatives from step 1 and step 2, we get: $$\int\left(e^{2 t}+2 \sqrt{t}\right) dt = \frac{1}{2}e^{2t} + \frac{4}{3}t^{3/2} + C$$ where \(C = C_1 + C_2\) is an arbitrary constant.

Step 4: Check the result by differentiation

We will now differentiate the antiderivative we found in step 3 and check if it results in the original integrand. $$\frac{d}{dt}\left(\frac{1}{2}e^{2t} + \frac{4}{3}t^{\frac{3}{2}} + C\right) = e^{2t} + 2t^{\frac{1}{2}} = e^{2t} + 2\sqrt{t}$$ The derivative of the antiderivative matches the original integrand, which confirms that our result is correct.

Key Concepts

Integration Techniques

Integration techniques are essential tools in calculus that help us solve various integral problems. One common method is the separation of a complex expression into simpler components. By doing this, we can tackle each part one at a time.
In our exercise, the integrand consists of two terms: \(e^{2t}\) and \(2\sqrt{t}\). We integrate each term separately. This approach simplifies solving indefinite integrals, which are integrals without given bounds.

• By breaking down complex expressions into simpler parts, integration becomes manageable.
• These techniques are vital when dealing with multiple terms in an integrand.

Using these techniques ensures accuracy and makes verifying the solution easier through differentiation.

Power Rule for Integration

The power rule for integration is a straightforward formula used to find the antiderivative of polynomial functions.
It states that for any real number \(n eq -1\), the integral of \(t^n\) is given by \(\int t^n \, dt = \frac{1}{n+1}t^{n+1} + C\), where \(C\) is the constant of integration.

• This rule is crucial for integrating terms with a variable raised to a power.
• In our example, \(2\sqrt{t}\) was first rewritten as \(2t^{\frac{1}{2}}\).

Applying the power rule here, we increased the exponent from \(\frac{1}{2}\) to \(\frac{3}{2}\), divided by the new exponent, and then multiplied by 2 to arrive at \(\frac{4}{3}t^{\frac{3}{2}}\).
Understanding this rule simplifies the integration process for polynomial-like terms.

Exponential Functions

Exponential functions are functions of the form \(e^{ax}\), where \(e\) is Euler's number (approximately 2.718), and \(a\) is a constant. The integration of exponential functions is unique because their derivative is proportional to the function itself.
For any constant \(a\), the rule becomes \(\int e^{ax} \, dx = \frac{1}{a}e^{ax} + C\). This is applied to integrate terms like \(e^{2t}\).

• Exponential integrals maintain the same exponential form as their antiderivatives.
• They are exceptionally useful in modeling growth patterns and decay in real-world problems.

In our exercise, using this integration rule directly results in \(\frac{1}{2}e^{2t}\). Understanding exponential functions and their integration rule is vital for a wide variety of mathematical contexts.