Find f. f '(t) = 20/1 + t² f(1) = 0

Solution:

Integrals are the values of the function found by the process of integration. 

The process of getting f(x) from f'(x) is called integration.

Given f '(t) = 20/ 1 + t² and f(1) = 0

We know that by definition of integral of tan⁻¹(x) = 1/1 + t²

f '(t) = 20 / 1 + t² = 20(1/ 1 + t²)

Integrate on both sides, we get

f(t) = 20tan⁻¹(t) + c

f(1) = 0; f(1) = 20tan⁻¹(1) + c = 0

c = -20(π/4) = -5π.

Therefore, f(x) = 20tan⁻¹(t) -5π.

Find f. f '(t) = 20/1 + t² f(1) = 0

Summary:

If f '(t) =20/1 + t² f(1) = 0, then f is f(x) = 20tan⁻¹(t) - 5π.