How can I find the value of this Maclaurin Series.

Question

Using the four non-zero terms of Maclaurin Series how can I find the value of $$\int_0^1 2x \cos ^2 x dx$$

When I solved I got Maclaurin Series as $2x-2x^3+\frac{2}{3}x^5$.

But I don't know how to apply the intervals.

Can anyone show how to do this.

Answer 1

First note that the first 4 (four) terms are $$2 x-2 x^3+\frac{2}{3} x^5 - \frac{4}{45} x^7$$ and the integral from this approximation is $\frac{3}{5}$. This is not too bad compared to the exact value $$\cos(1) \sin(1)+\frac{1}{2}\cos(1)^2 \approx 0.600612004276\approx 0.6 = \frac{3}{5}$$

Edit: An antiderivative of the four term expression is $$F(x) = x^2-\frac{1}{2}x^4+\frac{1}{9}x^6-\frac{1}{90}x^8$$ and therefore you get the integal as $F(1)-F(0) = \frac{3}{5} - 0$.

Answer 2

The right term should be approximation.

After you found the four non-zero terms,

$$2x\cos^2(x) \approx \sum_{i=1}^7 a_i x^i$$

Integrate it term by term.

$$\int_0^12x\cos^2(x)\, dx \approx \sum_{i=1}^7 a_i \int_0^1x^i\, dx$$

Answer 3

$I =\int_0^12x\cos^2(x)\,dx$

The MacLaurin series of $f(x) = 2x\cos^2(x) $ is $2x - 2x^3 +\frac23x^5-\frac4{45}x^7 + \frac2{315}x^9........$

$I \approx \int_0^12x - 2x^3 +\frac23x^5-\frac4{45}x^7 \,dx$

$I\approx x^2-\frac12x^4 +\frac19x^6-\frac1{90}x^8\bigg]_0^1$

$I\approx 1-\frac12+\frac19-\frac1{90}$

$I \approx \frac35$

NOTE: im not sure if MacLaurin series are the best way to solve this integral ,I'd much rather use integration by parts

$I = (\frac{2x (\sin(2x)+x)+\cos(2x)}4)\bigg]_0^1$

$I = \frac{2.(\sin(2)+1)+\cos(2)}{4}-\frac{\cos(0)}4$

$I=0.600612$