Question

Let f (x) = cos 3x. Cos 2x Determine the primitive function with F (Xo) = √2

Answer 1

0.5647

Explanation:

We have the following function:

`f(x)=\cos 3x\cdot\cos 2x`

The primitive function is the definite integral of the function, evaluated from 0 to 2.

Therefore, we calculate the integral and evaluate:

`\begin{gathered} \int ^(√(2))_0\mleft(\cos 3x\cdot\cos 2x\mright)dx \\ \text{ Let's rewrite the function as follows:} \\ \cos \: \: 3x\cdot\cos \: \: 2x=(\cos(3x+2x)+\cos(3x-2x))/(2) \\ \cos \: \: 3x\cdot\cos \: \: 2x=(1)/(2)\cdot(\cos 5x+\cos x) \\ \int ^{\sqrt[]{2}}_0((1)/(2)\cdot(\cos 5x+\cos x))dx=(1)/(2)\int ^{\sqrt[]{2}}_0\cos 5xdx+(1)/(2)\int ^{\sqrt[]{2}}_0\cos xdx \\ \text{ we integrate:} \\ \int ^{\sqrt[]{2}}_0\cos 5xdx=(1)/(5)\cdot\sin (5\sqrt[]{2})-(1)/(5)\cdot\sin (5\cdot0)=0.1417-0=0.1417 \\ \int ^{\sqrt[]{2}}_0\cos xdx=\sin (\sqrt[]{2})-\sin (0)=0.98776-0=0.98776 \\ \int ^{\sqrt[]{2}}_0(\cos 3x\cdot\cos 2x)dx=(1)/(2)(0.1417+0.98776) \\ \int ^{\sqrt[]{2}}_0(\cos 3x\cdot\cos 2x)dx=0.5647 \end{gathered}`

The result is 0.5647