which of the following is equivalent to ∫(2x³ + 1)²dx? a ∫(4x⁶ + 4x³ + 1)dx b ∫(2x³ + 1)dx·∫(2x³ + 1)dx c…

Explanation:

Step1: Expand the integrand

Use the formula ((a + b)^2=a^{2}+2ab + b^{2}). Here (a = 2x^{3}) and (b = 1), so ((2x^{3}+1)^{2}=(2x^{3})^{2}+2\times(2x^{3})\times1+1^{2}=4x^{6}+4x^{3}+1). Then (\int(2x^{3}+1)^{2}dx=\int(4x^{6}+4x^{3}+1)dx). Also, we can use substitution method. Let (u = 2x^{3}+1), then (du=6x^{2}dx), and (\int(2x^{3}+1)^{2}dx=\frac{1}{6x^{2}}\int u^{2}du) is incorrect as the substitution process is wrong. And (\int(2x^{3}+1)dx\cdot\int(2x^{3}+1)dx) is not equal to (\int(2x^{3}+1)^{2}dx) because ((\int f(x)dx)(\int f(x)dx)\neq\int(f(x))^{2}dx).

Step2: Analyze substitution - correct form

If (u = 2x^{3}+1), then (du = 6x^{2}dx), and (\int(2x^{3}+1)^{2}dx=\frac{1}{6}\int u^{2}\frac{du}{x^{2}}) (the given D has wrong coefficient). The correct substitution - based form is (\int u^{2}du) with (u = 2x^{3}+1) after proper handling of (dx) in terms of (du).

Answer:

C. (\int u^{2}du), where (u = 2x^{3}+1)