Question

Find a function f that has the given derivative f and value f(c). Find an antiderivative of f by hand, if possible; if it is not possible to anti differentiate by hand, use the Second Fundamental Theorem of Calculus to write down an antiderivative.

$f^{\text{'}}\left(x\right)=\sin \left(x^2\right);f\left(0\right)=0;$

Short answer

The function is$F\left(x\right)={\int }_a^x\sin \left(t^2\right)dt.$

Step-by-step solution

Step 1.Given information 

Given function is$f^{\text{'}}\left(x\right)=\sin \left(x^2\right);f\left(0\right)=0;$.

Step 2.The formula 

The antiderivative of f is $$\begin{gathered} F\left(x\right)={\int }_a^{x}f\left(t\right)dt \\ F\left(x\right)={\int }_a^x\sin \left(t^2\right)dtx\in [a,b] \end{gathered}$$

Also given that $f\left(0\right)=0$the antiderivative is,

$F\left(x\right)={\int }_0^x\sin \left(t^2\right)dt$ .

Step 3.The solution

The solution is$F\left(x\right)={\int }_a^x\sin \left(t^2\right)dt.$