Integration Using Trigonometric Substitution
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
Figure 2 shows a sketch of the curve with equation
\[y = \frac{16 \sin 2x}{(3 + 4 \sin x)^2} \quad 0 \leq x \leq \frac{\pi}{2}\]
The region \( R \), shown shaded in Figure 2, is bounded by the curve, the x-axis and the line with equation \( x = \frac{\pi}{6} \).
Using the substitution \( u = 3 + 4 \sin x \), show that the area of \( R \) can be written in the form \( a \sin b \), where \( a \) and \( b \) are rational constants to be found.
(7)
Solution: Setting Up the Integral and Evaluating the Integral
Key Concepts Used:
- Trigonometric Substitution: Using \( u = 3 + 4 \sin x \) to simplify the integrand.
- Double Angle Identity: Recognizing \( \sin 2x = 2 \sin x \cos x \).
- Limit Transformation: Changing integration limits to match the new variable.
- Basic Integration Rules:
\[\begin{align*}
\int \frac{1}{u} du &= \ln |u| \\
\int \frac{1}{u^2} du &= -\frac{1}{u}
\end{align*}\]
- Logarithmic Properties: Combining logarithmic terms.
- Exact Value Evaluation: Calculating definite integral values.
Step-by-Step Solution:
1. Apply the Substitution
Given substitution:
\[u = 3 + 4 \sin x\]
Compute differential:
\[\frac{du}{dx} = 4 \cos x\]
\[dx = \frac{du}{4 \cos x}\]
2. Rewrite the Integrand
Original integrand: \(\frac{16 \sin 2x}{(3 + 4 \sin x)^2}\)
Using double angle identity: \(\sin 2x = 2 \sin x \cos x\)
Substitute:
\[\frac{16 \times 2 \sin x \cos x}{u^2} \times \frac{du}{4 \cos x} = \frac{8 \sin x}{u^2} du\]
3. Express \(\sin x\) in Terms of \(u\)
From substitution: \(\sin x = \frac{u – 3}{4}\)
Thus:
\[\frac{8}{u^2} \times \frac{u – 3}{4} du = \frac{2(u – 3)}{u^2} du\]
4. Simplify the Integrand
Separate terms:
\[\frac{2u}{u^2} – \frac{6}{u^2} = \frac{2}{u} – \frac{6}{u^2}\]
5. Change the Limits
Original limits \(x = 0\) to \(x = \frac{\pi}{6}\):
When \(x = 0\): \(u = 3 + 4 \times 0 = 3\)
When \(x = \frac{\pi}{6}\): \(u = 3 + 4 \times \frac{1}{2} = 5\)
\[\int_{3}^{5} \left( \frac{2}{u} – \frac{6}{u^2} \right) du\]
6. Integrate the Simplified Expression
\[\int \left( \frac{2}{u} – \frac{6}{u^2} \right) du\]
Integrate term by term:
\[\int \left( \frac{2}{u} – \frac{6}{u^2} \right) du\]
\[\int \frac{2}{u} du = 2\ln|u|\]
\[\int -\frac{6}{u^2} du = \frac{6}{u}\]
\[2\ln|u| + \frac{6}{u} + C\]
7. Evaluate at Limits
At \( u = 7 \): \( 2\ln 7 + \frac{6}{7} \)
At \( u = 5 \): \( 2\ln 5 + \frac{6}{5} \)
Subtract lower limit from upper limit:
\[\left( 2\ln 7 + \frac{6}{7} \right) – \left( 2\ln 5 + \frac{6}{5} \right)\]
8. Simplify the Expression
Combine logarithmic terms:
\[2(\ln 7 – \ln 5) + \left( \frac{6}{7} – \frac{6}{5} \right)\]
Simplify fractions:
\[2\ln \left( \frac{7}{5} \right) + \left( \frac{30 – 42}{35} \right)\]
\[2\ln \left( \frac{7}{5} \right) – \frac{12}{35}\]
Convert logarithmic term:
\[\ln \left( \frac{49}{25} \right) – \frac{12}{35}\]
Final Answer:
\[\ln \left( \frac{49}{25} \right) – \frac{12}{35}\]
