How do you integrate #f(x)=x+1# using the power rule?

Question

Ezra Adams · Accepted Answer

#1/2x^2+x+c#

The #color(blue)"power rule for integration"# is.

#color(orange)"Reminder " color(red)(bar(ul(|color(white)(2/2)color(black)(intax^ndx=a/(n+1)x^(n+1))color(white)(2/2)|)))#

#rArrint(x+1)dx=int(x^1+x^0)dx#

#=1/2x^2+x^1+c=1/2x^2+x+c#

where c, is the constant of integration.

Ezekiel Alexander · Answer

undefined To integrate $ f(x) = x + 1 $ using the power rule, follow these steps:

1. Identify the function to integrate, $ f(x) = x + 1 $.
2. Apply the power rule, which states that the integral of $ x^n $ with respect to $ x $ is $ \frac{x^{n+1}}{n+1} + C $, where $ C $ is the constant of integration.
3. For $ f(x) = x + 1 $, integrate each term separately.
4. Integrate $ x $ using the power rule: $ \int x \, dx = \frac{x^2}{2} + C $.
5. Integrate $ 1 $ using the power rule: $ \int 1 \, dx = x + C $.
6. Combine the integrals: $ \int (x + 1) \, dx = \frac{x^2}{2} + x + C $.