Chapter 1: Q18MP (page 1)

Find the Maclaurin series of the following functions.
$arc t a n x = \int_{0}^{x} \frac{d u}{1 + u^{2}}$

Short Answer

Expert verified

Maclaurin series of $\arctan x is x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots . .$

Step by step solution

The Maclaurin series and the given function.

Function is $\arctan x = \int_{0}^{x} \frac{d u}{1 + u^{2}}$

The Maclaurin series of $\int_{0}^{x} \frac{d u}{1 + u^{2}}$ is given as follows:

$\int_{0}^{x} \frac{d u}{1 + u^{2}} = \int_{0}^{x} (1 - u^{2} + u^{4} - u^{6} + \dots . .) d u \int_{0}^{x} \frac{d u}{1 + u^{2}} = {(u - \frac{u^{3}}{3} + \frac{u^{5}}{5} - \frac{u^{7}}{7} + \dots . .)}_{0}^{x} \int_{0}^{x} \frac{d u}{1 + u^{2}} = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots$

Now, we have:

$\frac{1}{1 + x} = 1 - x + x^{2} - x^{3} + \dots . .$

Calculation by the Maclaurin series.

Substitute $u^{2}$ for $x$ in expansion of $\frac{1}{1 + x}$ as follows:

$\begin{array}{rcl} \frac{1}{1 + u^{2}} & = & 1 - u^{2} + {(u^{2})}^{2} - {(u^{2})}^{3} + \dots . . \\ \frac{1}{1 + u^{2}} & = & 1 - u^{2} + u^{4} - u^{6} + \dots . . \end{array}$

Apply integration for $\lim$ $0$ to $x$ in the above equation as follows:

Thus, the Maclaurin series of $\arctan x is x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots . .$

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Find the Maclaurin series of the following functions.
$arc t a n x = \int_{0}^{x} \frac{d u}{1 + u^{2}}$

Short Answer

Step by step solution

The Maclaurin series and the given function.

Calculation by the Maclaurin series.

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Find the Maclaurin series of the following functions.arctanx=∫0xdu1+u2

Short Answer

Step by step solution

The Maclaurin series and the given function.

Calculation by the Maclaurin series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Physics of Motion

Physical Quantities And Units

Measurements

Linear Momentum

Solid State Physics

Space Physics

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series of the following functions.
$arc t a n x = \int_{0}^{x} \frac{d u}{1 + u^{2}}$