Problem 93 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(x^{2}+2^{x}\right) d x $$

Step-by-Step Solution

Answer

\(\int \left(x^2 + 2^x\right) \, dx = \frac{x^{3}}{3} + \frac{2^{x}}{\ln 2} + C\)

Step 1: Identify the Parts of the Integral

The integral given is \(\int\left(x^{2}+2^{x}\right) d x\) which is a sum of two separate terms: \(x^2\) and \(2^x\). This means we can integrate each term independently and then add the results together.

Step 2: Integrate the Polynomial Term

Let's start by integrating the polynomial term \(x^2\). The formula for integrating \(x^n\) is \(\frac{x^{n+1}}{n+1} + C\). Applying this to \(x^2\), we get:\[\int x^{2} \;dx = \frac{x^{3}}{3} + C_1\]where \(C_1\) is a constant of integration.

Step 3: Integrate the Exponential Term

Next, we integrate the exponential term \(2^x\). The integral of \(a^x\) for a constant \(a\) is \(\frac{a^x}{\ln a} + C\). Applying this to \(2^x\), we have:\[\int 2^{x} \;dx = \frac{2^{x}}{\ln 2} + C_2\]where \(C_2\) is another constant of integration.

Step 4: Combine the Integrals

Now, we combine the results from Step 2 and Step 3 to get the final solution. Add the integrals together:\[\int \left(x^2 + 2^x\right) \; dx = \frac{x^{3}}{3} + \frac{2^{x}}{\ln 2} + C\]where \(C = C_1 + C_2\) is the combined constant of integration.

Key Concepts

Polynomial Integration

When we talk about polynomial integration, we refer to the process of integrating polynomial expressions. Polynomials are expressions involving terms like \(x^n\), where \(n\) is a non-negative integer. The integral of a polynomial term \(x^n\) is given by the formula \(\frac{x^{n+1}}{n+1} + C\), where \(C\) is a constant of integration.

• Each term in a polynomial can be integrated independently because integration is a linear operation.
• The formula tells us to raise the power of \(x\) by one and divide by this new exponent.
• Remember to always add a constant, denoted here as \(C\), to represent the indefinite nature of the integral.

In our exercise, we integrated \(x^2\) by applying this rule, resulting in \(\frac{x^3}{3} + C_1\). It's crucial to differentiate each constant of integration for each term before final combination.

Exponential Integration

Exponential integration deals with integrating expressions where the variable is in the exponent. A common form is \(a^x\), where \(a\) is a constant base.

• To integrate \(a^x\), use the formula \(\frac{a^x}{\ln a} + C\).
• It's necessary to divide by \(\ln a\) (the natural logarithm of the base) to normalize the expression.
• This formula allows us to handle exponential growth or decay scenarios.

In our example, for the exponential term \(2^x\), we applied \(\int 2^x \, dx = \frac{2^x}{\ln 2} + C_2\). Notice the use of a different constant of integration \(C_2\) here.

Constants of Integration

When finding indefinite integrals, constants of integration are key to capturing all possible values of the antiderivative. Indefinite integrals do not have specific limits of integration, hence there's an infinite number of antiderivatives.

• Each integration term gets its own constant initially, like \(C_1\) and \(C_2\) in our solution.
• These constants combine into a single constant \(C\) in the final integrated expression.
• The general form for an indefinite integral is \(F(x) + C\), where \(F(x)\) is any antiderivative of the function.

Constants of integration ensure that all possible vertical shifts of the antiderivative curve are accounted for, reflecting each potential solution the original differential equation might yield.