Problem 37 - Calculus for Biology and Medicine

Question

Find the general antiderivative of the given function. $$ f(x)=x^{-7}+3 x^{5}+\sin (2 x) $$

Step-by-Step Solution

Answer

The general antiderivative is \( F(x) = -\frac{1}{6}x^{-6} + \frac{1}{2}x^6 - \frac{1}{2}\cos(2x) + C \).

Step 1: Identify the Antiderivative Rules

To find the general antiderivative, we need to apply the basic integration rules for powers of x and trigonometric functions. The integration rule for a power function \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) applies when \( n eq -1 \). For trigonometric functions, \( \int \sin(ax) \, dx = -\frac{1}{a}\cos(ax) + C \) is used, where \( a \) is a constant.

Step 2: Apply the Power Rule to Each Term

Let's apply the power rule individually to each term in the function.1. For \( x^{-7} \), the antiderivative is \( \frac{x^{-7+1}}{-7+1} = \frac{x^{-6}}{-6} = -\frac{1}{6}x^{-6} \).2. For \( 3x^{5} \), the antiderivative is \( 3 \cdot \frac{x^{5+1}}{5+1} = 3 \cdot \frac{x^6}{6} = \frac{1}{2}x^6 \).

Step 3: Apply the Sine Integration Rule

For the trigonometric function \( \sin(2x) \), we use the formula \( \int \sin(ax) \, dx = -\frac{1}{a}\cos(ax) \):- The antiderivative of \( \sin(2x) \) is \( -\frac{1}{2}\cos(2x) \).

Step 4: Combine the Antiderivatives and Add the Constant

Combine all the antiderivative results and include the constant of integration \( C \):- The general antiderivative of \( f(x) = x^{-7} + 3x^5 + \sin(2x) \) is: \[ F(x) = -\frac{1}{6}x^{-6} + \frac{1}{2}x^6 - \frac{1}{2}\cos(2x) + C \]

Key Concepts

Integration Rules

To solve integrals, understanding integration rules is essential. These rules originate from the concept of antiderivatives, which are functions that represent the original function before differentiation. An essential integration rule is for power functions, expressed as \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). Each function type has its associated integration rule.

• For power functions, you increase the exponent by one and divide by the new exponent.
• For trigonometric functions, specific rules apply based on the function, like sine and cosine rules.

Knowing the correct rule helps you navigate through problems effectively. Once you apply the proper integration rule to each term, combine the resulting antiderivatives, always remembering to add a constant of integration \( C \) at the end, since integration can have infinitely many solutions.

Power Rule

The power rule is a simple but very powerful tool when dealing with polynomials or any variable raised to a power, which we often encounter in algebra and calculus. In the context of integration, it's essential to correctly apply the power rule by:

• Increasing the exponent by 1.
• Dividing by the new exponent.

This leads to the formula \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). In cases where your function is a sum of multiple power terms, apply the rule to each term separately.

In our specific example, for \( x^{-7} \), increase \(-7\) by 1 to get \(-6\), resulting in \(-\frac{1}{6}x^{-6}\), and for \(3x^5\), increase \(5\) by 1, giving you \(\frac{1}{2}x^6\). These steps are systematic and can be repeated as needed for any polynomial terms.

Trigonometric Integration

Trigonometric functions like sine and cosine possess particular integration formulas that make integration simpler. When facing functions such as \( \sin(2x) \), the formula \( \int \sin(ax) \, dx = -\frac{1}{a}\cos(ax) + C \) becomes crucial. This formula accounts for the change in variable during integration.

Applying trigonometric integration can be simplified by identifying any constants inside the function:

• Replace \(a\) with the constant from within the function. In our example, \(a = 2\), so the integral of \( \sin(2x) \) is \(-\frac{1}{2}\cos(2x)\).

Combining trigonometric integrals with power rule results helps to assemble the complete antiderivative of the given function. Such an operation reveals how each part of the original function connects within the final answer. Integration constants remind us of the infinite possibilities derived from integration.