beautyfulgift9378 beautyfulgift9378
  • 02-08-2017
  • Mathematics
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Question part points submissions used use a power series to approximate the definite integral, i, to six decimal places. 0.3 ln(1 + x4) dx 0

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LammettHash
LammettHash LammettHash
  • 02-08-2017
[tex]\ln(1+x)=\displaystyle-\sum_{n=1}^\infty\frac{(-x)^n}n[/tex]

[tex]\implies\ln(1+x^4)=\displaystyle-\sum_{n=1}^\infty\frac{(-x^4)^n}n[/tex]

[tex]\displaystyle-\int_0^{0.3}\sum_{n=1}^\infty\frac{(-x^4)^n}n\,\mathrm dx[/tex]
[tex]=\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}n\int_0^{0.3}x^{4n}\,\mathrm dx[/tex]
[tex]=\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}n\dfrac{x^{4n+1}}{4n+1}\bigg|_0^{0.3}[/tex]
[tex]=\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}(0.3)^{4n+1}}{n(4n+1)}[/tex]
[tex]\approx0.000485[/tex]
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