Q. 31 - Calculus

Question

Use the Second Fundamental Theorem of Calculus to write down three antiderivatives of each function f in Exercises 31–34.

$f\left(x\right)={\sin }^2\left(3^x\right)$

Step-by-Step Solution

Answer

The three antiderivatives for the function $f\left(x\right)={\sin }^2\left(3^x\right)$ are $F\left(x\right)={\int }_{-1}^x{\sin }^2\left(3t\right)dt, F\left(x\right)={\int }_1^x{\sin }^2\left(3t\right)dt, F\left(x\right)={\int }_2^x{\sin }^2\left(3t\right)dt$.

Step 1. Given Information.

The function:

$f\left(x\right)={\sin }^2\left(3^x\right)$

Step 2. Graph the function.

Graph the function.

Step 3. Find the anti-derivatives.

If F is an anti-derivative of f, f is continuous on [a,b], then $F\left(x\right)={\int }_a^{x}f\left(t\right).dt$ for all $x\in [a,b]$.

So, $F\left(x\right)={\int }_{-1}^x{\sin }^2\left(3t\right)dt$ is defined to be an anti-derivative of the given function.

Similarly, the other two anti-derivatives are:

$F\left(x\right)={\int }_1^x{\sin }^2\left(3t\right)dt, F\left(x\right)={\int }_2^x{\sin }^2\left(3t\right)dt$.