Findthe derivative with respect to x of the integral from 2 to x to the 4th power of the tangent of x squared, dx

Question

Find



Answer 1

The derivative and integral act as opposites to each other:

d/dx ∫ (a to x) f(x) dx = d/dx( F(x) - F(a)), where F(x) is whatever function the integral spits out (antiderivative of f(x)) and a is any real number

d/dx (F(x)) = f(x) (They cancel each other out, in effect).

d/dx (F(a)) = 0 (Since a is some arbitrary number, F(a) will just be some arbitrary real number, assuming it is defined. The derivative of any real number is 0).

Therefore d/dx ∫ (a to x) f(x) dx = f(x) - 0 = f(x)

In our case, since tan⁴(2²) = tan⁴(4) is defined, d/dx ∫ (2 to x) tan⁴(x²) dx = tan⁴(x²).