(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 integral. Consider the integral

∫e^ax * sin(nx) dx

where a and n are both positive constants.

(a) Use a power-law expanision for e^ax 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:

∫₀^0.1e^5xsin(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 x_min = 0 and x_max = 0.1 with a = 5 and n = 3.

(d) If you use Mathematica or integral tables, you should find that:

∫e^axsin(nx)dx = (e^ax / 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.