Question

Fill in each of the blanks:

(a) $\int x^{6}dx=\underline{}+C.$

(b) is an antiderivative of role="math" localid="1648619282178" $x^6.$

(c) The derivative of is $x^6.$

Short answer

(a) $\int x^{6}dx=\underline{\frac{x^7}{7}}+C.$

(b)$\underline{\frac{x^7}{7}}$ is an antiderivative of $x^6.$

(c) The derivative of $\underline{\frac{x^7}{7}}$is$x^6.$

Step-by-step solution

Step 1. Given information. 

The given incomplete statements are following.

(a) $\int x^{6}dx=\underline{}+C.$

(b) is an antiderivative of $x^6.$

(c) The derivative of is $x^6.$

Step 1. Part (a). 

integral of $x^6$

$$\begin{gathered} \int x^{6}dx=\left(\frac{x^{6+1}}{6+1}\right)+C \\ =\frac{x^7}{7}+C \end{gathered}$$

so $\int x^{6}dx=\underline{\frac{x^7}{7}}+C.$

Step 3. Part (b). 

As $\int x^{6}dx=\frac{x^7}{7}+C.$

So $\underline{\frac{x^7}{7}}$is an antiderivative of$x^6.$

Step 4. Part (c) 

antiderivative of $x^6.$

role="math" localid="1648620114580" $$\begin{gathered} \int x^{6}dx=\left(\frac{x^{6+1}}{6+1}\right)+C \\ =\frac{x^7}{7}+C \end{gathered}$$

so The derivative of $\underline{\frac{x^7}{7}}$is$x^6.$