Chapter 8: Q. 32 (page 704)

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

Short Answer

Expert verified

The power series can be given as

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

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" $\int \cos (x^{3}) d x = x - \frac{x^{7}}{14} + \frac{x^{13}}{13 \cdot 24} - \frac{x^{19}}{19 \cdot 720} + . . . . . \int_{0}^{1} \cos (x) d x = 1 - \frac{1^{7}}{14} + \frac{1^{13}}{13 \cdot 24} - \frac{1^{19}}{19 \cdot 720} + . . . . .$

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

Short Answer

Step by step solution

Given information

Find the power series of cos(x3)

Find the remaining power series

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

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