The **constant rule of integration** tells you how to find an integral for a constant quantity like 7, ⅓ or π. The rule is defined as:

∫

adx=ax.

## Constant Rule of Integration Examples

**Example problem #1**: Use the constant rule of integration to evaluate the indefinite integral y = ∫ 4 *dx*.

Step 1: **Place the constant in the question into the rule**:

∫*4* *dx* → 4*x*.

Step 2: **Add a “+ C”** (*Why add a + C”?*):

The solution is 4*x* + C.

**Example problem #2**: Evaluate the indefinite integral ∫ ½ *dx*.

Step 1: **Place the constant into the rule:**

*½* *dx* → *½*.

Step 2: **Add a “+ C”**:

The solution is *½x* + C.

**Example problem #3**: Evaluate the following:

Step 1: **Place the constant into the rule**:

= (6/π)*x*.

Step 2: **Add a “+ C”:**

The solution is = (6/π)*x* + C.

Notice that in the above problem π is a constant, so you can use the constant rule of integration. Euler’s number e is also a constant, so you can use this rule. However, e^{x} is not a constant because of the x. However, the integral of e^{x} is itself: e^{x} + C.

## Constant Multiple Rule

The **constant multiple rule** for integrals is similar, but this time we’re concerned with finding an integral for a function multiplied by a constant:

The integral of ∫*c* *f*(*x*) *dx* is *c* ∫*f*(*x*) *dx*.

They are almost exactly the same: the constant is brought out in front of the integral. In the case of the constant rule of integration though, that process only leaves “dx”, which is converted to x in our integral. With the constant multiple rule there is still a function f(x) to evaluate *after* you’ve used the constant multiple rule.

**Example problem #4: **Evaluate ∫5*x dx.*

Step 1: **Pull the constant out in front:**

5 ∫*x dx.*

Step 2: Integrate x with the power rule and simplify:

Step 3: **Add a + C**: