A vanishing cubic term does not make an integral exact

In this question, you may use expansions from the List of Formulae (MF26).

(a) Given that f(x)=esin⁡3xf(x)=e^{\sin3x}, find the Maclaurin expansion of f(x)f(x) in ascending powers of xx, up to and including the term x2x^2. Show that this expansion is independent of x3x^3. marks

(b) Use your expansion from part (a) and integration to find an approximate expression for ∫esin⁡3xx2 dx\int\frac{e^{\sin3x}}{x^2}\,dx. Hence obtain an approximate value for ∫0.10.2(2x)2esin⁡3x dx\int_{0.1}^{0.2}(\frac2x)^2e^{\sin3x}\,dx, correct to 4 decimal places. marks

(c) Use your calculator to find ∫0.10.2(2x)2esin⁡3x dx\int_{0.1}^{0.2}(\frac2x)^2e^{\sin3x}\,dx, correct to 4 decimal places. mark

(d) A student compares the answers to parts (b) and (c) and concludes that the approximation is accurate. Give a reason to support his conclusion. mark

[2023/DHS/Prelim/P1/6]

Total marks: 9

Solution

(a)

Composition of series

Find the Maclaurin expansion of esin⁡3xe^{\sin3x} up to x2x^2 and show it has no x3x^3 term.

Explore this idea

Hint

  1. sin⁡3x=3x−27x36+⋯=3x−92x3+⋯\sin3x=3x-\frac{27x^3}6+\cdots=3x-\frac92x^3+\cdots.

  2. Substitute into eu=1+u+u22+u36e^u=1+u+\frac{u^2}2+\frac{u^3}6.

Answer · (a)

esin⁡3x=1+3x+9x2/2+O(x4)e^{\sin3x}=1+3x+9x^2/2+O(x^4); cubic coefficient 0.

Explore the idea

e^(sin 3x) and its series

Compare the function with a finite polynomial. Change the highest included power and test x, then use the error plot to see how the approximation changes at that x.

Highest included powerxⁿ, n = 2
Test value xx = 0.15
Try an example
Function
1.54491
P₂(x)
1.55125
Error
0.00634

The shading is a local comparison window, not a convergence or accuracy guarantee.

Teal marks the function, amber marks the polynomial, and the red segment joins their values at the selected x.

Does keeping more powers help at this x?

Highest included power n runs horizontally; signed error Pₙ(x) − f(x) runs vertically. Zero means agreement at this x. A missing power leaves the polynomial unchanged; individual steps need not improve the error.

  • Function
  • Series truncated at the chosen power
  • A local comparison window, not a convergence or accuracy guarantee
Key idea

The cubic coefficient is 0, so degrees 2 and 3 give the same polynomial.

Ideas to try
Compare degree 2 and degree 3: the polynomial does not change.
(b)

Integrating a series

Use your expansion to find an approximate expression for ∫esin⁡3xx2 dx\int\dfrac{e^{\sin3x}}{x^2}\,dx, and hence ∫0.10.2(2x)2esin⁡3xdx\int_{0.1}^{0.2}\left(\dfrac2x\right)^2e^{\sin3x}dx to 4 d.p.

Explore this idea

Hint

  1. Divide the series by x2x^2.

  2. Integrate term by term.

  3. Multiply by 4 and use the limits.

Answer · (b)

∫f(x)/x2 dx≈−1/x+3ln⁡∣x∣+9x/2+C\int f(x)/x^2\,dx\approx-1/x+3\ln|x|+9x/2+C; definite estimate 30.117830.1178.

Explore the idea

True area against the series estimate

Drag x to compare the numerical signed integral with the series estimate. Their difference shows the approximation error.

Upper limit xx = 0.20
Reset the controls

How the shaded areas add up

The limits run left to right. Area above the axis adds; area below the axis subtracts.

Above-axis contribution
29.999
Below-axis contribution
0
Signed integral
29.999

All three bars share one scale. Positive contributions extend right of zero; negative contributions extend left.

Signed integral from 0.10 to x
29.999
series antiderivative − F(0.1)
30.118
Estimate − integral
0.118
0.39% of the integral’s magnitude
Compare the running integral on a graph
  • Integrand
  • Numerical signed integral from the start to x
  • Part of the integrand above the axis
  • Part of the integrand below the axis
  • Series estimate of the signed integral; it need not agree with the true integral
Key idea

Truncating the series makes the integral slightly too large: 30.1178 against the true 29.9995.

Ideas to try
Drag xx to 0.2: the dashed (series) value is 30.1178 and the solid (true) area is 29.9995.
(c)

Numerical integration

Use your calculator to find ∫0.10.2(2x)2esin⁡3xdx\int_{0.1}^{0.2}\left(\dfrac2x\right)^2e^{\sin3x}dx to 4 d.p.

Hint

  1. Use the calculator's definite integral in radian mode.

Answer · (c)

29.999529.9995 (4 d.p.).

(d)

Judging an approximation

A student concludes the approximation is accurate. Give a reason.

Hint

  1. Compute the percentage error.

  2. Note the interval is close to 0.

Answer · (d)

Relative error ≈0.3943%\approx0.3943\%; endpoints close to 0 and first omitted numerator term quartic.