Q. 32 - Calculus

Question

Use the Maclaurin series for cos x to find series representations for $\cos \left(x^3\right),\int \cos \left(x^3\right)dx, {\int }_0^1\cos \left(x^3\right)dx$

Step-by-Step Solution

Answer

The power series can be given as 

$$\begin{gathered} \cos \left(x^3\right)=1-\frac{x^6}{2}+\frac{x^{12}}{24}-\frac{x^{18}}{720}+..... \\ \int \cos \left(x^3\right)dx=x-\frac{x^7}{14}+\frac{x^{13}}{13·24}-\frac{x^{19}}{19·720}+..... \\ {\int }_0^1\cos \left(x^3\right)dx=1-\frac{1^7}{14}+\frac{1^{13}}{13·24}-\frac{1^{19}}{19·720}+..... \end{gathered}$$

Step 1: Given information

We are given a power series of cos (x)

Step 2: Find the power series of cos ( x 3 )

We are given power series of cos(t) which is

$$\begin{gathered} \cos \left(t\right)=1-\frac{t^2}{2}+\frac{t^4}{4!}-\frac{t^6}{6!}+.... \\ replace t=x^3 \\ \cos \left(x^3\right)=1-\frac{x^6}{2}+\frac{x^{12}}{4!}-\frac{x^{18}}{6!}+.... \end{gathered}$$

Step 3: Find the remaining power series

Now we integrate the power series of $\cos \left(x^3\right)$

we get,

$$\begin{gathered} \int \cos \left(x^3\right)dx=x-\frac{x^7}{14}+\frac{x^{13}}{13·24}-\frac{x^{19}}{19·720}+..... \\ {\int }_0^1\cos \left(x\right)dx=1-\frac{1^7}{14}+\frac{1^{13}}{13·24}-\frac{1^{19}}{19·720}+..... \end{gathered}$$