Introduction

Integration is one of the two main operations in calculus, the other being differentiation. Integrals allow us to calculate areas, volumes, accumulated quantities, and more. To simplify integration, mathematicians have developed a set of integration rules. These rules make it easier to work with different types of functions and solve integrals efficiently. In this article, we will cover the fundamental integration rules and provide examples to illustrate how each rule is applied.

Basic Integration Rules

\[\begin{align*} \textbf{1. Constant Rule:} \quad \int a \, dx = ax + C \end{align*}\]

The constant rule states that the integral of a constant \( a \) is simply the constant multiplied by \( x \), plus a constant of integration \( C \). For example:

\[\begin{align*} \int 5 \, dx = 5x + C \end{align*}\]

\[\begin{align*} \textbf{2. Power Rule:} \quad \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad (n \neq -1) \end{align*}\]

The power rule is used when integrating functions of the form \( x^n \). Increase the exponent by one and divide by the new exponent. For example:

\[\begin{align*} \int x^3 \, dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C \end{align*}\]

\[\begin{align*} \textbf{3. Sum Rule:} \quad \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \end{align*}\]

The sum rule states that the integral of a sum is the sum of the integrals. For example:

\[\begin{align*} \int (x^2 + 3x) \, dx = \int x^2 \, dx + \int 3x \, dx = \frac{x^3}{3} + \frac{3x^2}{2} + C \end{align*}\]

\[\begin{align*} \textbf{4. Difference Rule:} \quad \int (f(x) – g(x)) \, dx = \int f(x) \, dx – \int g(x) \, dx \end{align*}\]

Similar to the sum rule, the difference rule states that the integral of a difference is the difference of the integrals. For example:

\[\begin{align*} \int (x^3 – 2x) \, dx = \int x^3 \, dx – \int 2x \, dx = \frac{x^4}{4} – x^2 + C \end{align*}\]

\[\begin{align*} \textbf{5. Constant Multiple Rule:} \quad \int a \cdot f(x) \, dx = a \cdot \int f(x) \, dx \end{align*}\]

The constant multiple rule allows us to pull out constants from the integral. For example:

\[\begin{align*} \int 3x^2 \, dx = 3 \cdot \int x^2 \, dx = 3 \cdot \frac{x^3}{3} + C = x^3 + C \end{align*}\]

\[\begin{align*} \textbf{6. Exponential Rule:} \quad \int e^x \, dx = e^x + C \end{align*}\]

The exponential rule states that the integral of \( e^x \) is \( e^x \) itself, plus the constant of integration \( C \). For example:

\[\begin{align*} \int e^x \, dx = e^x + C \end{align*}\]

Special Integration Rules

\[\begin{align*} \textbf{1. Integral of \( \frac{1}{x} \):} \quad \int \frac{1}{x} \, dx = \ln|x| + C \end{align*}\]

The integral of $\dfrac1x$ is quite a common integral so it is good to keep in mind.

\[\begin{align*} \textbf{2. Trigonometric Functions:} \end{align*}\]

There are several integration rules for basic trigonometric functions:

\(\displaystyle \int \sin(x) \, dx = -\cos(x) + C\)
\(\displaystyle \int \cos(x) \, dx = \sin(x) + C\)
$\displaystyle \int \tan(x) \, dx = -\ln|cos(x)| + C$
\(\displaystyle \int \sec^2(x) \, dx = \tan(x) + C\)
\(\displaystyle \int \csc^2(x) \, dx = -\cot(x) + C\)
\(\displaystyle \int \sec(x) \tan(x) \, dx = \sec(x) + C\)
\(\displaystyle \int \csc(x) \cot(x) \, dx = -\csc(x) + C\)

Practice Problems

Let’s look at a few examples to see how these rules can be applied in practice.

1. Evaluate \(\displaystyle \int (3x^2 + 4x) \, dx \):
   Using the sum rule, power rule, and constant multiple rule:
   \[\begin{align*} \int (3x^2 + 4x) \, dx = 3 \int x^2 \, dx + 4 \int x \, dx \end{align*}\]
   \[\begin{align*} = 3 \cdot \frac{x^3}{3} + 4 \cdot \frac{x^2}{2} + C = x^3 + 2x^2 + C \end{align*}\]
2. Evaluate \(\displaystyle \int e^{x} + \frac{1}{x} \, dx \):
   Using the exponential rule and the integral of \( \dfrac{1}{x} \):
   \[\begin{align*} \int \left( e^{x} + \frac{1}{x} \right) \, dx = \int e^x \, dx + \int \frac{1}{x} \, dx \end{align*}\]
   \[\begin{align*} = e^x + \ln|x| + C \end{align*}\]
3. Evaluate \(\displaystyle \int (2 \cos(x) – 5 \sin(x)) \, dx \):
   Using the constant multiple rule and trigonometric integration rules:
   \[\begin{align*} \int (2 \cos(x) – 5 \sin(x)) \, dx = 2 \int \cos(x) \, dx – 5 \int \sin(x) \, dx \end{align*}\]
   \[\begin{align*} = 2 \sin(x) + 5 \cos(x) + C \end{align*}\]

Conclusion

Integration rules are essential tools in calculus, allowing us to find anti-derivatives and calculate areas under curves. By mastering these rules, we can simplify complex integrals and solve a wide range of mathematical problems. Whether you are dealing with basic functions, polynomials, exponentials, or trigonometric functions, these rules provide a reliable framework for approaching integration.