Q. 54 - Calculus

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.

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

Step-by-Step Solution

Answer

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

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

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