Power Rule for Integration

Definition

The Power Rule for Integration states that for any real number $n \neq -1$, the integral of $x^n$ with respect to $x$ is given by $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $C$ is the constant of integration. This rule simplifies the process of finding antiderivatives for polynomial functions, allowing for straightforward calculations and applications in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. The Power Rule applies only when $n \neq -1$. For $n = -1$, the integral becomes a natural logarithm: $$\int x^{-1} \, dx = \ln |x| + C$$.
2. This rule is particularly useful in calculating integrals of polynomials, as it allows for quick determination of antiderivatives without needing to rely on more complex methods.
3. When applying the Power Rule, remember to increase the exponent by 1 and then divide by that new exponent.
4. The Power Rule can be used iteratively, meaning you can apply it multiple times to find antiderivatives of higher-degree polynomials.
5. In practice, using the Power Rule for Integration helps in solving problems related to areas, volumes, and other applications of calculus.

Review Questions

• How does the Power Rule for Integration simplify finding antiderivatives of polynomial functions?
  • The Power Rule for Integration simplifies finding antiderivatives by providing a straightforward formula that applies to polynomial functions. Instead of deriving complicated expressions, one can simply apply the rule to increase the exponent by one and divide by this new exponent. This streamlined approach allows students and practitioners to quickly and efficiently calculate antiderivatives for various polynomial expressions, making it a fundamental tool in calculus.
• Compare the application of the Power Rule for Integration when $n = 2$ and when $n = -1$. What are the differences in approach?
  • When applying the Power Rule for Integration with $n = 2$, we use the formula to get $$\int x^2 \, dx = \frac{x^{3}}{3} + C$$. However, when $n = -1$, we cannot use this rule directly because it leads to division by zero. Instead, we must recognize that $$\int x^{-1} \, dx = \ln |x| + C$$ represents a different approach entirely, showing how special cases must be handled separately when integrating.
• Evaluate the integral $$\int (3x^4 - 5x^2 + 4) \, dx$$ using the Power Rule for Integration and explain each step.
  • To evaluate $$\int (3x^4 - 5x^2 + 4) \, dx$$ using the Power Rule, we apply the rule to each term individually. For $3x^4$, we increase the exponent to 5 and divide by 5: $$3 \cdot \frac{x^{5}}{5} = \frac{3}{5} x^{5}$$. For $-5x^2$, we do similarly: $$-5 \cdot \frac{x^{3}}{3} = -\frac{5}{3} x^{3}$$. Lastly, for the constant 4, it integrates to $4x$. Combining these results gives us: $$\frac{3}{5} x^{5} - \frac{5}{3} x^{3} + 4x + C$$. Each step follows from applying the Power Rule directly and showcases how to integrate polynomial expressions efficiently.

Related terms

Antiderivative: An antiderivative of a function is another function whose derivative equals the original function, essentially reversing the process of differentiation.

Definite Integral: A definite integral calculates the net area under a curve between two specific limits, providing a numerical value rather than a function.

Constant of Integration: The constant of integration, denoted as $C$, represents an arbitrary constant added to an antiderivative to account for the family of functions that differ by a constant.