Q. 7 - Calculus

Question

Fill in each of the blanks:

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

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

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

Step-by-Step Solution

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 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 $x^6.$

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

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