There are many different ways to compute integrals. While all of the integration rules are important, they all have their place and can be used in different situations. When it comes to integrating polynomials, the inverse power rule is the most useful technique. The inverse power rule states:
\[\begin{align*} \boxed{\int x^n \: dx = \dfrac{x^{n+1}}{n+1} + C}, \end{align*}\]
where $C$ is the constant of integration.
Approaching Integrals of Polynomials
The standard process when integrating polynomials is to first write the polynomial as the sum of its terms and then integrate each term independently. We are able to do this due to the sum and difference rules of integration. We can then use the constant multiple rule and factor out the coefficient of each term. After that, we can apply the inverse power rule.
Example 1 – Standard Polynomial
Evaluate
\[\begin{align*} \int (3x^2 + 4x – 9) \: dx \end{align*}\]
We can proceed by first splitting up the integral using the sum/difference rule, and then integrating each term separately using the power rule. We will also need to apply the constant multiple rule to factor out the constants from our integrals before applying the power rule:
\[\begin{align*} \int (3x^2 + 4x – 9) \: dx &= \int 3x^2 \: dx + \int 4x \: dx – \int 9 \: dx\\ &= 3 \cdot \int x^2 \: dx + 4 \cdot \int x \: dx – \int 9 \: dx\\ &= 3 \cdot \dfrac{x^3}{3} + 4 \cdot \dfrac{x^2}{2} – 9x + C\\ &= \boxed{x^3 + 2x^2 – 9x + C} \end{align*}\]
Example 2 – Variable in the Denominator
Evaluate
\[\begin{align*} \int \dfrac{3}{x^4} \: dx \end{align*}\]
When it comes to integrating fractions with $x$ in the denominator, it is helpful to convert the variable into the form $x^a$, where $a$ is negative and then use the inverse power rule. In this case, we may rewrite $\dfrac{3}{x^4}$ as $3 \cdot x^{-4}$. We now apply the inverse power rule. We rewrite the integral as
\[\begin{align*} \int \dfrac{3}{x^4} \: dx &= \int 3 \cdot x^{-4} \: dx\\ &= 3 \cdot \int x^{-4} \: dx\\ &= 3 \cdot \dfrac{x^{-3}}{-3} + C\\ &= -x^{-3} + C\\ &= \boxed{-\dfrac1{x^3} + C} \end{align*}\]
Example 3 – Fractional Exponents
Evaluate
\[\begin{align*} \int \sqrt[3]{x^5} \: dx \end{align*}\]
Here, again, in order to use the inverse power rule, we must rewrite the expression in the form $x^a$. First, notice that $\sqrt[3]{x^5} = x^{\frac53}$. Even though the exponent is a fraction, the inverse power rule applies the same way. We will proceed as before, carefully combining fractions in the denominator and noting that division by a fraction is equivalent to multiplication by a reciprocal:
\[\begin{align*} \int \sqrt[3]{x^5} \: dx &= \int x^{\frac53} \: dx\\ &= \dfrac{x^{\frac53 + 1}}{\frac53 + 1} + C\\ &=\dfrac{x^{\frac83}}{\frac83} + C\\ &=x^{\frac83}\cdot\dfrac{3}{8} + C\\ &=\boxed{\dfrac{3\sqrt[3]{x^8}}{8} + C} \end{align*}\]
Example 4 – Irrational Exponents
Evaluate
\[\begin{align*} \int x^{\pi} \: dx \end{align*}\]
While this looks strange, remember that $\pi$ is just another number. We may apply the inverse power rule as follows:
\[\begin{align*} \int x^{\pi} \: dx &= \boxed{\dfrac{x^{\pi + 1}}{\pi + 1} + C} \end{align*}\]
We cannot simplify further, hence, we keep our answer as is.
Conclusion
If there are no special functions, like exponentials, logarithms or trig functions, the inverse power rule can very often be sufficient to calculate the integral. There are many cases when other techniques are needed. For example, integrating expressions like $\dfrac{1}{x^2 + 3x +2}$ may require partial fractions, even though there are no special functions. However, the inverse power rule should always be your first go-to if there are no special functions. The power rule is a key tool to keep in mind when working with all sorts of integrals as even when integrating by parts or using $u$-substitution, the power rule tends to make a comeback!
