Q. 6 - Calculus

Question

Fill in each of the blanks:

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

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

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

Step-by-Step Solution

Answer

(a) $\int \underline{6x^5} dx=x^6+C.$

(b) $x^6$is an antiderivative of $\underline{6x^5}.$

(c) The derivative of $x^6$is $\underline{6x^5}.$

Step 1. Given information.

 The given incomplete statements are following.

a) $\int \underline{ } dx=x^6+C.$

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

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

Step 1. Part (a).

derivative of $x^6.$

$$\begin{gathered} \int 6x^5 dx=6\left(\frac{x^{5+1}}{5+1}\right)+C \\ =x^6+C \\ =x^6 \end{gathered}$$

so $\int \underline{6x^5} dx=x^6+C.$

Step 3. Part (b).

As $\int 6x^{5}dx=x^6+C.$

So $x^{6 }$is an antiderivative of $6x^5$.

Step 4. Part (c)

Derivative of $x^6.$

$$\begin{gathered} \frac{d}{dx}{x}^6=6x^{6-1} \\ =6x^5 \end{gathered}$$

So derivative of $x^6$is $\underline{6x^5}.$