Question

final milestone | calculus i
final milestone
calculate $intleft(48sqrt3{x^{2}} + 23 - \frac{21}{x} + \frac{25}{x^{6}}
ight)dx$
$\bigcirc$ $\frac{144}{5}x^{\frac{5}{3}} + 23x - 21ln|x| - \frac{5}{x^{5}} + c$
$\bigcirc$ $80x^{\frac{5}{3}} + 23x - 21ln|x| + 25ln|x^{6}| + c$
$\bigcirc$ $\frac{144}{5}x^{\frac{5}{3}} + 23x - 21ln|x| + 25ln|x^{6}| + c$
$\bigcirc$ $\frac{144}{5}x^{\frac{5}{3}} + 23x - 21ln|x| - \frac{5}{x^{5}}$

Explanation:

Step1: Rewrite radicals/reciprocals

Rewrite the integrand using exponents:
$$48x^{\frac{2}{3}} + 23 - 21x^{-1} + 25x^{-6}$$

Step2: Integrate term-by-term

Apply power rule $\int x^n dx = \frac{x^{n+1}}{n+1}$ ($n
eq-1$) and $\int \frac{1}{x}dx=\ln|x|$:

• $\int 48x^{\frac{2}{3}}dx = 48\cdot\frac{x^{\frac{2}{3}+1}}{\frac{2}{3}+1}=48\cdot\frac{3}{5}x^{\frac{5}{3}}=\frac{144}{5}x^{\frac{5}{3}}$
• $\int 23dx = 23x$
• $\int -21x^{-1}dx = -21\ln|x|$
• $\int 25x^{-6}dx = 25\cdot\frac{x^{-6+1}}{-6+1}=25\cdot\frac{x^{-5}}{-5}=-\frac{5}{x^5}$

Step3: Add constant of integration

Combine terms and add $+c$ (indefinite integral constant).

Answer:

$\frac{144}{5}x^{\frac{5}{3}} + 23x - 21\ln|x| - \frac{5}{x^5} + c$ (matches the first option)