integral of u du
Understand the Problem
The question is asking to find the integral of the function u du, which means we need to perform an integration operation on the expression u multiplied by du.
Answer
$$ \int u \, du = \frac{u^2}{2} + C $$
Answer for screen readers
$$ \int u , du = \frac{u^2}{2} + C $$
Steps to Solve
- Set up the integral
We need to set up the integral to find the area under the curve represented by the function. The function we are integrating is $u$ with respect to $u$. We write it as:
$$ \int u , du $$
- Apply the power rule of integration
The power rule states that the integral of $u^n$ with respect to $u$ is given by:
$$ \int u^n , du = \frac{u^{n+1}}{n+1} + C $$
for $n \neq -1$. Since our function is simply $u$ (which can be expressed as $u^1$), we can apply the power rule:
$$ \int u^1 , du = \frac{u^{1+1}}{1+1} + C = \frac{u^2}{2} + C $$
- Write the final answer
After applying the power rule, we obtain the final result of the integral:
$$ \int u , du = \frac{u^2}{2} + C $$
$$ \int u , du = \frac{u^2}{2} + C $$
More Information
The result represents the area under the curve of the function $u$ from a given point to another. The constant $C$ is added because the integral represents a family of functions that differ by a constant.
Tips
- Forgetting to add the constant of integration $C$ is a common mistake. Always remember to include it when performing indefinite integrals.
