Find F ′(x) for F of x equals the integral from x cubed to 1 of the cosine of t raised to the 4th power, dt.

Question

Find F ′(x) for .

 A. cos(1) – cos(x¹²) 

B. –3x²cos(x¹²) 

C. cos(x⁷) 

D. –cos(x¹²)

Answer 1

Use integral properties to switch the bounds of integration to make the requested integral = - ∫₁^{x^3} cos(t⁴)dt.

Next, use the FTC to evaluate: - (3x²)cos((x³)⁴) = –3x²cos(x¹²) .

Note that the 3x² term comes from the derivative of the upper bound function, while the x¹² arises from replacing t in the integrand with the upper bound.