d. Is your solution to part c an overestimate or underestimate of the actual number of bacteria colonies at 1:30? Justify your answer using concepts from calculus.

Evaluate the indefinite integral. \( \int 8 \sin ^{3}(x) \cos (x) d x=\square+C \)

Evaluate the definite integral. \( \int_{0}^{1} \sqrt{1 x+3} d x \)

Evaluate the integral by making the appropriate substitution: \( u=\square \frac{d x}{(2 x+2)^{4}} \), \( \int \frac{d x}{(2 x+2)^{4}}=\square \)

Evaluate the definite integral. \( \int_{0}^{\frac{\pi}{4}} \sin (4 t) d t \)

Evaluate the indefinite integral. \( \int \frac{5}{(t+3)^{2}} d t \) \( \square+C \)