Q. 32 Use the Maclaurin series for cos... [FREE SOLUTION]

Chapter 8: Q. 32 (page 704)

Use the Maclaurin series for cos x to find series representations for $\cos (x^{3}), 鈭� \cos (x^{3}) d x, {鈭�}_{0}^{1} \cos (x^{3}) d x$

Short Answer

Expert verified

The power series can be given as

$\cos (x^{3}) = 1 - \frac{x^{6}}{2} + \frac{x^{12}}{24} - \frac{x^{18}}{720} + . . . . . 鈭� \cos (x^{3}) d x = x - \frac{x^{7}}{14} + \frac{x^{13}}{13 路 24} - \frac{x^{19}}{19 路 720} + . . . . . {鈭�}_{0}^{1} \cos (x^{3}) d x = 1 - \frac{1^{7}}{14} + \frac{1^{13}}{13 路 24} - \frac{1^{19}}{19 路 720} + . . . . .$

Step by step solution

Given information

We are given a power series of cos (x)

Find the power series of cos(x3)

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

$\cos (t) = 1 - \frac{t^{2}}{2} + \frac{t^{4}}{4!} - \frac{t^{6}}{6!} + . . . . r e p l a c e t = x^{3} \cos (x^{3}) = 1 - \frac{x^{6}}{2} + \frac{x^{12}}{4!} - \frac{x^{18}}{6!} + . . . .$

Find the remaining power series

Now we integrate the power series of $\cos (x^{3})$

we get,

role="math" localid="1654232162397" $鈭� \cos (x^{3}) d x = x - \frac{x^{7}}{14} + \frac{x^{13}}{13 路 24} - \frac{x^{19}}{19 路 720} + . . . . . {鈭�}_{0}^{1} \cos (x) d x = 1 - \frac{1^{7}}{14} + \frac{1^{13}}{13 路 24} - \frac{1^{19}}{19 路 720} + . . . . .$

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91影视

Use the Maclaurin series for cos x to find series representations for $\cos (x^{3}), 鈭� \cos (x^{3}) d x, {鈭�}_{0}^{1} \cos (x^{3}) d x$

Short Answer

Step by step solution

Given information

Find the power series of cos(x3)

Find the remaining power series

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91影视

Use the Maclaurin series for cos x to find series representations forcos(x3),鈭�cos(x3)dx,鈭�01cos(x3)dx

Short Answer

Step by step solution

Given information

Find the power series of cos(x3)

Find the remaining power series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Discrete Mathematics

Geometry

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use the Maclaurin series for cos x to find series representations for $\cos (x^{3}), 鈭� \cos (x^{3}) d x, {鈭�}_{0}^{1} \cos (x^{3}) d x$