match each indefinite integral to its result, where y > 0 and c is a constant. integral result ∫(12y² / (4y³…

Explanation:

Step1: Recall substitution rule

For $\int\frac{f'(y)}{f(y)}dy=\ln|f(y)| + C$.

Step2: Analyze $\int\frac{12y^{2}}{4y^{3}+1}dy$

Let $u = 4y^{3}+1$, then $du=12y^{2}dy$. So $\int\frac{12y^{2}}{4y^{3}+1}dy=\int\frac{du}{u}=\ln|u|+C=\ln(4y^{3}+1)+C$. This does not match any of the given results.

Step3: Analyze $\int\frac{20y^{3}}{5y^{4}+1}dy$

Let $u = 5y^{4}+1$, then $du = 20y^{3}dy$. So $\int\frac{20y^{3}}{5y^{4}+1}dy=\int\frac{du}{u}=\ln|u|+C=\ln(5y^{4}+1)+C$.

Step4: Analyze $\int\frac{5y^{4}+1}{y^{5}+y}dy$

Let $u=y^{5}+y$, then $du=(5y^{4}+1)dy$. So $\int\frac{5y^{4}+1}{y^{5}+y}dy=\int\frac{du}{u}=\ln|u|+C=\ln(y^{5}+y)+C$.

Answer:

$\int\frac{12y^{2}}{4y^{3}+1}dy$ - No match $\int\frac{20y^{3}}{5y^{4}+1}dy$ - $\ln(5y^{4}+1)+C$ $\int\frac{5y^{4}+1}{y^{5}+y}dy$ - $\ln(y^{5}+y)+C$