Question

homework9: problem 6
(1 point)
find the most general antiderivative of ( f(x)=10 - 10x^{3}-10x^{5}+10x^{7} ).
note: any arbitrary constants used must be an upper - case \c\.
( f(x)= )

Explanation:

Step1: Recall the power rule for antiderivatives

The power rule for antiderivatives states that the antiderivative of \( x^n \) (where \( n
eq -1 \)) is \( \frac{x^{n + 1}}{n+1} \), and the antiderivative of a constant \( k \) is \( kx \). We will apply this rule term - by - term to the function \( f(x)=10 - 10x^{3}-10x^{5}+10x^{7} \).

Step2: Find the antiderivative of the first term

For the term \( 10 \) (which can be written as \( 10x^{0} \) since \( x^{0}=1 \) for \( x
eq0 \)), using the power rule with \( n = 0 \), the antiderivative is \( 10\times\frac{x^{0 + 1}}{0+1}=10x \).

Step3: Find the antiderivative of the second term

For the term \( - 10x^{3} \), using the power rule with \( n = 3 \), the antiderivative is \( - 10\times\frac{x^{3+1}}{3 + 1}=-10\times\frac{x^{4}}{4}=-\frac{10}{4}x^{4}=-\frac{5}{2}x^{4} \).

Step4: Find the antiderivative of the third term

For the term \( - 10x^{5} \), using the power rule with \( n = 5 \), the antiderivative is \( - 10\times\frac{x^{5+1}}{5 + 1}=-10\times\frac{x^{6}}{6}=-\frac{10}{6}x^{6}=-\frac{5}{3}x^{6} \).

Step5: Find the antiderivative of the fourth term

For the term \( 10x^{7} \), using the power rule with \( n = 7 \), the antiderivative is \( 10\times\frac{x^{7+1}}{7 + 1}=10\times\frac{x^{8}}{8}=\frac{10}{8}x^{8}=\frac{5}{4}x^{8} \).

Step6: Add the arbitrary constant

We also need to add an arbitrary constant \( C \) (since the most general antiderivative includes all possible antiderivatives, which differ by a constant).

Combining all the antiderivatives of the individual terms, we get:

\( F(x)=10x-\frac{5}{2}x^{4}-\frac{5}{3}x^{6}+\frac{5}{4}x^{8}+C \)

Answer:

\( 10x-\frac{5}{2}x^{4}-\frac{5}{3}x^{6}+\frac{5}{4}x^{8}+C \)