What is the integral of 7x?
Understand the Problem
The question is asking for the integral of the function 7x with respect to x. This is a straightforward calculus problem where we will apply the power rule for integration.
Answer
The integral of $7x$ is $\frac{7}{2} x^2 + C$.
Answer for screen readers
The integral of the function $7x$ with respect to $x$ is
$$ \frac{7}{2} x^2 + C $$
Steps to Solve
- Identify the Function to Integrate
We need to integrate the function $f(x) = 7x$ with respect to $x$.
- Apply the Power Rule for Integration
The power rule states that for any function of the form $x^n$, the integral is given by:
$$ \int x^n , dx = \frac{x^{n+1}}{n+1} + C $$
In our case, $7x$ can be rewritten as $7x^1$.
- Integrate using the Power Rule
Now apply the power rule to $7x^1$:
$$ \int 7x^1 , dx = 7 \cdot \frac{x^{1+1}}{1+1} + C $$
This simplifies to:
$$ = 7 \cdot \frac{x^2}{2} + C $$
- Simplify the Expression
Now, simplify the expression:
$$ = \frac{7}{2} x^2 + C $$
Where $C$ is the constant of integration.
The integral of the function $7x$ with respect to $x$ is
$$ \frac{7}{2} x^2 + C $$
More Information
This result is derived from the application of the power rule in integration, which is a fundamental concept in calculus. The constant of integration $C$ represents any constant value that can be added to the function, since the integral represents a family of functions.
Tips
- Forgetting to add the constant of integration $C$ at the end.
- Incorrectly applying the power rule by not properly adjusting the exponent.
- Miscalculating the coefficients during the integration process.
