Mathematics, 29.05.2020 04:58 bg988763p7cl2d
(a) Evaluate the integral ∫2048x2+4dx. Your answer should be in the form kπ, where k is an integer. What is the value of k? (Hint: darctan(x)dx=1x2+1 ) k= 6 (b) Now, lets evaluate the same integral using power series. First, find the power series for the function f(x)=48x2+4. Then, integrate it from 0 to 2, and call it S. S should be an infinite series ∑[infinity]n=0an . What are the first few terms of S?
Answers: 3
![=\int\limits^2_0 {\frac{48}{x^2+2} } \, dx \\\\=\frac{48}{2} [\tan ^-^1(\frac{\pi}{2}) ]^2_0](/tpl/images/0669/8583/5d782.png)
![=24[\tan ^-^1(1)- \tan^-^1(0)]\\\\=24(\frac{\pi}{4} )\\\\=6\pi\\\\ k \pi=6 \pi\\\\k=6](/tpl/images/0669/8583/cf217.png)
in equation (1)
![=[{12x-x^3+\frac{3}{20} x^5-\frac{3}{16}\frac{x^7}{7} +\frac{3}{64} \frac{x^9}{9} -\frac{3}{256}\frac{x^1^1}{11} +...]^2_0 }](/tpl/images/0669/8583/b7f12.png)
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(a) Evaluate the integral ∫2048x2+4dx. Your answer should be in the form kπ, where k is an integer....
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