Understanding the Power Rule for Integration
The power rule for integration is a fundamental technique in calculus used to find the antiderivative of functions of the form x^n, where n is a constant. The rule states that the integral of x^n dx equals (x^{n+1})/(n+1) + C, for n ≠ -1. This reverses the power rule for differentiation, allowing quick computation of indefinite integrals for polynomial terms. To apply it, identify the power n, increase the exponent by 1, divide by the new exponent, and add the constant of integration C.
Key Steps and Principles
Start by rewriting the integrand in power form if needed. For example, constants can be factored out. The rule applies directly to monomials but extends to polynomials by integrating term by term. Remember, for n = -1, the integral is ln|x| + C, which is a special case not covered by the basic power rule. Always include +C for indefinite integrals to account for the family of antiderivatives. This principle ensures accuracy in solving differential equations and area problems.
Practical Example
Consider integrating ∫ (3x^4 - 2x^2 + 5) dx. Apply the power rule to each term: for 3x^4, it's 3*(x^5)/5 = (3/5)x^5; for -2x^2, it's -2*(x^3)/3 = (-2/3)x^3; for 5 (which is 5x^0), it's 5x. Adding C gives (3/5)x^5 - (2/3)x^3 + 5x + C. Verify by differentiating: the result returns the original function, confirming correctness.
Importance and Real-World Applications
The power rule is essential for evaluating definite integrals in physics, engineering, and economics, such as calculating displacement from velocity or total cost from marginal cost. It forms the basis for more advanced integration techniques like substitution or parts. Mastering it builds confidence in handling complex integrals and avoids common errors in calculus applications, like forgetting the +C or mishandling exponents.