Question

Use the Second Fundamental Theorem of Calculus to find F (x). F(x) = integral sqroot t⁴ + 5 dt between the limits - 1 and x F(x)=

Answer 1

The Second Fundamental Theorem of Calculus states that the derivative of a function that is the integral of another function is equal to the original function. To find F(x), differentiate the given function and integrate it back to obtain F(x) = (1/5) * t^5 + 5t + C, where C is the constant of integration.

Step-by-step explanation:

The Second Fundamental Theorem of Calculus states that if the function F(x) is the integral of f(t) with respect to t from a constant a to x, then the derivative of F(x) with respect to x is equal to f(x).

In this case, F(x) = ∫(t⁴+5) dt from -1 to x. To find F(x), we first need to find f(x) by differentiating t⁴+5 with respect to t. The derivative of t⁴+5 is 4t³.

Therefore, F(x) = ∫(t⁴+5) dt = (1/5) * t⁵ + 5t + C, where C is the constant of integration.