(1 point) Calculate the following antiderivatives:
(a)
(b)
©

Ask by Gordon Reed. in the United States

Mar 30,2025

Question

(1 point) Calculate the following antiderivatives: (a) $$ \int x^{5} d x=\square+C $$ (b) $$ \int x^{8 / 9} d x=\square+C $$ (c) $$ \int x^{-6} \sqrt{x} d x=\square $$

UpStudy AI · Accepted Answer

**(a)**  
We have  
$$
\int x^5\,dx = \frac{x^{5+1}}{5+1}+C = \frac{x^6}{6}+C.
$$

**(b)**  
We have  
$$
\int x^{8/9}\,dx = \frac{x^{8/9+1}}{8/9+1}+C.
$$  
Notice that  
$$
8/9+1=\frac{8}{9}+\frac{9}{9}=\frac{17}{9}.
$$  
Thus,  
$$
\int x^{8/9}\,dx = \frac{x^{17/9}}{17/9}+C = \frac{9}{17}x^{17/9}+C.
$$

**(c)**  
First, rewrite the integrand:  
$$
x^{-6}\sqrt{x} = x^{-6}x^{1/2} = x^{-6+1/2} = x^{-11/2}.
$$  
Now, integrate:  
$$
\int x^{-11/2}\,dx = \frac{x^{-11/2+1}}{-11/2+1}+C.
$$  
Simplify the exponent in the numerator:  
$$
-11/2+1 = -11/2+\frac{2}{2} = -9/2.
$$  
Thus,  
$$
\int x^{-11/2}\,dx = \frac{x^{-9/2}}{-9/2}+C = -\frac{2}{9}x^{-9/2}+C.
$$

Therefore, the solutions are:

(a) $$\displaystyle \frac{x^6}{6}+C$$

(b) $$\displaystyle \frac{9}{17}x^{17/9}+C$$

(c) $$\displaystyle -\frac{2}{9}x^{-9/2}+C$$
(a) $$ \frac{x^6}{6} + C $$ (b) $$ \frac{9}{17}x^{17/9} + C $$ (c) $$ -\frac{2}{9}x^{-9/2} + C $$

(1 point) Calculate the following antiderivatives:
(a)
(b)
©

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