find the most general antiderivative or indefinite integral. ∫(4t³ + t/6) dt ∫(4t³ + t/6) dt = □

Explanation:

Step1: Apply sum - rule of integration

$\int(4t^{3}+\frac{t}{6})dt=\int4t^{3}dt+\int\frac{t}{6}dt$

Step2: Use power - rule for $\int4t^{3}dt$

The power - rule for integration is $\int t^{n}dt=\frac{t^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int4t^{3}dt$, since $\int at^{n}dt=a\int t^{n}dt$ ($a$ is a constant), we have $4\int t^{3}dt=4\times\frac{t^{3 + 1}}{3+1}=t^{4}$

Step3: Use power - rule for $\int\frac{t}{6}dt$

$\int\frac{t}{6}dt=\frac{1}{6}\int tdt$. By the power - rule, $\frac{1}{6}\times\frac{t^{1+1}}{1 + 1}=\frac{t^{2}}{12}$

Step4: Add the results and the constant of integration

$\int(4t^{3}+\frac{t}{6})dt=t^{4}+\frac{t^{2}}{12}+C$

Answer:

$t^{4}+\frac{t^{2}}{12}+C$