Integration Rules

The Essentials

There are certain universal properties of integrals that can make it easier to manipulate and solve problems involving integrals. These different properties are summarized below:

1. When taking an integral, the function inside the integral (the integrand) can be in terms of any variable. The variable inside the integrand has no effect on the process required to integrate it:

\[ \int f(x) \, dx = \int f(t) \, dt = \int f(u) \, du \]

2. When a function is multiplied by a constant, the constant can be pulled out in front of the integral:

\[ \int c \cdot f(x) \, dx = c \int f(x) \, dx \]

3. Sum Rule: The integral of a sum is equal to the sum of the integrals.

\[ \int f(x) + g(x) \, dx = \int f(x) \, dx + \int g(x) \, dx \]

4. Difference Rule: The integral of a difference is equal to the difference of the integrals.

\[ \int f(x) - g(x) \, dx = \int f(x) \, dx - \int g(x) \, dx \]

5. For definite integrals, when the bounds are the same, the integral will be equal to zero:

\[ \int_{a}^{a} f(x) \, dx = 0 \]

6. The bounds of the integral can be swapped, but we must then change the sign of the integral.

\[ \int_{a}^{b} f(x) \, dx = - \int_{b}^{a} f(x) \, dx \]

7. The bounds of an integral can be split up into multiple integrals, given that f(x) is integrable over the largest interval.

\[ \int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx \]

8. The average value of the function f(x) can be found using this formula.

\[ \text{f}_\text{ave} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx \]

The Table of Integrals found in the back of the Openstax textbook (referenced at the bottom of the page) is a helpful resource for more integral properties.

A Deeper Dive

Many of the Integral Rules listed above are straightforward enough that they don’t require a proof. However some of the rules could be explored further.

Proof 1

\[ \int_{a}^{a} f(x) \, dx = 0 \]

When we take an integral of a function, we are finding the area under its curve. In a case where the bounds are equal, we are evaluating the function two times at the same point. Think of this area as a change in \(y\) multiplied by a change in \(x\). When the bounds are equal, the change in \(x\) is zero which will make the area zero.

\[ \int_{a}^{a} f(x) \, dx = F(x) \bigg|_{a}^{a} = F(a) - F(a) = 0 \]

Proof 2

\[ \int_{a}^{b} f(x) \, dx = - \int_{b}^{a} f(x) \, dx \]

For a definite integral, the limit definition is:

\[ \int_{a}^{b} f(x) \, dx = \lim_{{n \to \infty}} \sum_{i=1}^{n} f(x_i) \Delta x \quad \Delta x = \frac{{b - a}}{n} \]

If we swap the bounds of the integral to be [b,a], the only thing that will change will be the \( \Delta x \) .

\[ \int_{b}^{a} f(x) \, dx = \lim_{{n \to \infty}} \sum_{i=1}^{n} f(x_i) \Delta x \quad \Delta x = \frac{{a - b}}{n} \]

When we make this change to \( \Delta x \) , it only changes the sign of \( \Delta x \) . This is why we get a sign change when we swap the bounds of a definite integral.

Proof 3

\[ \int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx \]

It is fairly simple to imagine that if b is between a and c, then the integral can be split into two pieces and then added together. One interesting idea though is that b could be outside the interval [a, c] and as long as f(x) is continuous over the largest interval, the formula is still true. Try it!

Practice

Problems:

1: Rewrite the integral to pull out a constant.

\[ \int 10t^2 \, dt \]

2: Rewrite the following integral as two separate integrals.

\[ \int_{1}^{4} x^2 + \sin(x) \, dx \]

3: Rewrite the following integral as two separate integrals.

\[ \int \cos(\theta) - \tan^2(\theta) \, d\theta \]

4: Evaluate the following integral.

\[ \int_{3}^{3} \sqrt{x^2 \tan^2(x) + 5x} \, dx \]

5: Find the average value of the following function over the interval [1,3].

\[ f(x) = 3x^2 \]

Solution:

1:

\[ 10 \int t^2 \, dt \]

2:

\[ \int_{1}^{4} x^2 \, dx + \int_{1}^{4} \sin(x) \, dx \]

3:

\[ \int \cos(\theta) \, d\theta - \int \tan^2(\theta) \, d\theta \]

4:

\[ \int_{3}^{3} \sqrt{x^2\tan^2(x) + 5x} \, dx = 0 \]

5:

\[ f_{\text{ave}} = \frac{1}{3 - 1} \int_{1}^{3} 3x^2 \, dx = 13 \]

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