Question

How do you use the Fundamental Theorem of Calculus to find the derivative of `int (cos(t^4) + t) dt` from -4 to sinx?

Answer 1

Use the Chain Rule along with the Fundamental Theorem of Calculus to get the answer `(cos(sin^{4}(x))+sin(x))*cos(x)`

Explanation:

For a function `F(x)=int_{a}^{x}f(t)\ dt`, the Fundamental Theorem of Calculus implies that, if `f` is continuous, then `F'(x)=f(x)`.

If `f(t)=cos(t^{4})+t`, the function to differentiate in this problem is `h(x)=int_{-4}^{sin(x)}f(t)\ dt`. If `F(x)=int_{-4}^{x}f(t)\ dt`, then `h(x)=F(sin(x))`.

By the Chain Rule, `h'(x)=F'(sin(x)) * d/dx(sin(x))`

`=(cos(sin^{4}(x))+sin(x)) * cos(x)`.

(In general, the Chain Rule says that, under appropriate assumptions about differentiability, `d/dx(f(g(x)))=f'(g(x))*g'(x)`).