(a) Use the power-law expansion for eox and sin(nx) near x = 0 to approximate the integrand as a cubic polynomial. (Retain all terms involving xº, x', x², and x³ in the integrand). (b) Integrate the cubic polynomial you obtained in part (a) to come up with an approximate expression for the integral above. Your answer should be a fourth-order polynomial in x. (c) Use your answer to part (b) to numerically estimate: 0.1 Je* sin(3x)dx (You may use a calculator, if you wish, to evaluate this. It should be just a plug and chug into your answer for part (b) using the limits xmin = 0 and xa = 0.1 with a =5 and max n = 3. d) Ify use Mathematica or integral tables, you should find that : ar fe“ sin(nx)dx = (asin(nx)– ncos(nx)) a' +n notice that this is written as an indefinite integral). Use this formula (and a calculator, if desired), to evaluate the definite integral in part (c). Compare to your answer calculated in part (c) from your power-law approximation to the integral.
Need help with this problem: Use the power law series to evaluate a relatively ugly
∫eax * sin(nx) dx
where a and n are both positive constants.
(a) Use a power-law expanision for eax and sin(nx) near x = 0 to approximate the integrand as a cubic polynomial. (Retain all terms involving x0, x1, x2, and x3 in the integrand).
(b) Integrate the cubic polynomial you obtained in part (a) to come up with an approximate expression for the integral above. Your answer should be a fourth-order polynomial in x.
(c) Use your answer to part (b) to numerically estimate:
∫00.1e5xsin(3x)dx
(You may use a calculator, if you wish, to evaluate this. It should be just a plug and chug into your answer for part (b) using the limits xmin = 0 and xmax = 0.1 with a = 5 and n = 3.
(d) If you use Mathematica or integral tables, you should find that:
∫eaxsin(nx)dx = (eax / a2 + n2)(a sin(nx) - n cos(nx))
(notice that this is written as an indefinite integral). Use this formula (and a calculator, if desired), to evaluate the definite integral in part (c). Compare to your answer calculated in part (c) from your power-law approximation to the integral.
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