# How do you use a power series to approximate # int_0^1xtan^-1xdx #?

##### 2 Answers

#### Explanation:

Let

The power series for

And so we get the integrand by multiplying by

And so replacing the integrand with the power series we have:

Integrating each term gives us:

Applying the limits gives us:

We will need to take sufficient terms to get an answer that is stable to 4dp, which I will perform in Excel as follows;

Due to the oscillating natures of the power series it takes a significant number of terms to get a stable answer, but we can conclude that

#### Explanation:

let

or

then the integral reduces to

now we use the integration by parts

so the integral=

=

now the indefinite integral

so for the limit we get the value of first integral=

similarly the indefinite integral

so for the req. limit we ge

so the ans. is =

=