find the antiderivative for each function when c equals 0. check your answers by differentiation. (a) g(x)=…

Explanation:

Step1: Recall power - rule for antiderivatives

The antiderivative of $x^n$ is $\frac{x^{n + 1}}{n+1}+C$ for $n\neq - 1$.

Step2: Find antiderivative of $g(x)=-7x^{-8}$

Let $n=-8$. Then $G(x)=\frac{-7x^{-8 + 1}}{-8 + 1}=\frac{-7x^{-7}}{-7}=x^{-7}$.

Step3: Check the answer for $g(x)$ by differentiation

Using the power - rule for differentiation $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $n=-7$, $\frac{d}{dx}(x^{-7})=-7x^{-7-1}=-7x^{-8}$.

Step4: Find antiderivative of $h(x)=x^{-8}$

Let $n = - 8$. Then $H(x)=\frac{x^{-8 + 1}}{-8+1}=-\frac{1}{7}x^{-7}$.

Step5: Check the answer for $h(x)$ by differentiation

Using the power - rule for differentiation, $\frac{d}{dx}(-\frac{1}{7}x^{-7})=(-\frac{1}{7})\times(-7)x^{-7 - 1}=x^{-8}$.

Step6: Find antiderivative of $k(x)=x^{-8}+3x^{2}+6$

Using the sum - rule for antiderivatives $\int(f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$. For $x^{-8}$, its antiderivative is $-\frac{1}{7}x^{-7}$; for $3x^{2}$, its antiderivative is $\frac{3x^{2 + 1}}{2+1}=x^{3}$; for $6$, its antiderivative is $6x$. So $K(x)=-\frac{1}{7}x^{-7}+x^{3}+6x$.

Step7: Check the answer for $k(x)$ by differentiation

$\frac{d}{dx}(-\frac{1}{7}x^{-7}+x^{3}+6x)=(-\frac{1}{7})\times(-7)x^{-8}+3x^{2}+6=x^{-8}+3x^{2}+6$.

Answer:

(a) $x^{-7}$ (b) $-\frac{1}{7}x^{-7}$ (c) $-\frac{1}{7}x^{-7}+x^{3}+6x$