# Science:Math Exam Resources/Courses/MATH103/April 2010/Question 05 (b)

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MATH103 April 2010
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### Question 05 (b)

In this problem, you are asked to find a value for the integral

${\displaystyle I=\int _{0}^{1}x\cos(x)\,dx}$

in two different ways.

(b) Use the Taylor series for the function ${\displaystyle \cos(x)}$ to write down a Taylor Series approximation for ${\displaystyle x\cos(x)}$ and find an approximation to the integral ${\displaystyle I}$ using the first 3 terms of that series.

(Leave your answer in terms of the three fractions, i.e. you need not compute a simplified fraction nor a decimal approximation. Some factorial values that may be useful: ${\displaystyle 2!=2}$, ${\displaystyle 3!=6}$, ${\displaystyle 4!=24}$, ${\displaystyle 5!=120}$, ${\displaystyle 6!=720}$, ${\displaystyle 7!=5040}$.

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