To find the antiderivative of the function [tex]\( f(x) = 13x^{12} \)[/tex], we follow these steps:
1. Identify the general rule for integration:
The integral of [tex]\( x^n \)[/tex] is [tex]\( \frac{x^{n+1}}{n+1} \)[/tex], where [tex]\( n \neq -1 \)[/tex].
2. Apply the rule to our specific function:
Here, [tex]\( n = 12 \)[/tex]. We increase the exponent by 1 and then divide by the new exponent.
[tex]\[
\int 13x^{12} \, dx = 13 \cdot \int x^{12} \, dx = 13 \cdot \left( \frac{x^{12+1}}{12+1} \right)
\][/tex]
3. Calculate the integral:
[tex]\[
= 13 \cdot \frac{x^{13}}{13}
\][/tex]
4. Simplify the expression:
[tex]\[
= x^{13}
\][/tex]
5. Include the constant of integration:
The general form of the indefinite integral (antiderivative) includes a constant [tex]\( C \)[/tex].
[tex]\[
F(x) = x^{13} + C
\][/tex]
To check our work by differentiating the antiderivative:
6. Differentiate [tex]\( F(x) \)[/tex] using the power rule for differentiation, which states that [tex]\( \frac{d}{dx}\left(x^n\right) = n \cdot x^{n-1} \)[/tex]:
[tex]\[
\frac{d}{dx} \left( x^{13} + C \right) = \frac{d}{dx} \left( x^{13} \right) + \frac{d}{dx}(C)
\][/tex]
7. Evaluate the derivatives:
[tex]\[
= 13x^{12} + 0 = 13x^{12}
\][/tex]
Since the derivative of [tex]\( x^{13} + C \)[/tex] is [tex]\( 13x^{12} \)[/tex], which is the original function [tex]\( f(x) \)[/tex], our antiderivative is confirmed to be correct.
Thus, the antiderivatives of [tex]\( f(x) = 13x^{12} \)[/tex] are:
[tex]\[
F(x) = x^{13} + C
\][/tex]
