High School

Which of the following is equivalent to \(\int (2x^3 + 1)^2 \, dx\)?

1) \(\int (4x^6 + 4x^3 + 1) \, dx\)

2) \(\int (4x^6 + 2x^3 + 1) \, dx\)

3) \(\int (4x^6 + 2x^3) \, dx\)

4) \(\int (4x^6 + 4x^3) \, dx\)

Answer :

Final answer:

The integral expression (2x^3+1)^2 dx is equivalent to ˄«(4x^6+4x^3+1) dx, which is option 1 from the choices given.

Explanation:

The expression (2x^3+1)^2 can be expanded using the algebraic identity (a+b)^2 = a^2 + 2ab + b^2. When we apply this identity to the given expression, we get:

  • (2x^3)^2 + 2(2x^3)(1) + (1)^2
  • 4x^6 + 4x^3 + 1

Therefore, the integral expression ˄«(2x^3+1)^2 dx is equivalent to ˄«(4x^6+4x^3+1) dx, which matches option 1) from the provided choices.

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