Find all the antiderivatives of the following function. Check your work by taking the derivative.

[tex]\[ f(x) = 13 x^{12} \][/tex]

The antiderivatives of [tex]\( f(x) = 13 x^{12} \)[/tex] are [tex]\( F(x) = \square \)[/tex].



Answer :

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]