Key Concepts and Formulas

• Trigonometric Substitution: Integrals of the form $\int \frac{(\sin \theta \pm \cos \theta)}{a+b \sin 2\theta} d\theta$ can often be solved by substituting $t = \sin \theta \mp \cos \theta$.
• Identity for $\sin 2\theta$: The square of the substitution variable is related to $\sin 2\theta$: $(\sin \theta - \cos \theta)^2 = 1 - \sin 2\theta$ and $(\sin \theta + \cos \theta)^2 = 1 + \sin 2\theta$. This allows us to express $\sin 2\theta$ in terms of $t$.
• Standard Integral Form: The integral $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a+x}{a-x} \right| + C$.

Step-by-Step Solution

Let the given integral be $I$. $I = 80 \int\limits_0^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta$

Step 1: Identify the substitution and its derivative. The numerator is $(\sin \theta + \cos \theta) d\theta$. This suggests the substitution $t = \sin \theta - \cos \theta$. Differentiating $t$ with respect to $\theta$: $dt = (\cos \theta - (-\sin \theta)) d\theta$ $dt = (\cos \theta + \sin \theta) d\theta$ This perfectly matches the numerator of the integrand.

Step 2: Express $\sin 2\theta$ in terms of the substitution variable $t$. We know that $t = \sin \theta - \cos \theta$. Squaring both sides: $t^2 = (\sin \theta - \cos \theta)^2$ $t^2 = \sin^2 \theta + \cos^2 \theta - 2 \sin \theta \cos \theta$ Using the identities $\sin^2 \theta + \cos^2 \theta = 1$ and $\sin 2\theta = 2 \sin \theta \cos \theta$: $t^2 = 1 - \sin 2\theta$ Rearranging to solve for $\sin 2\theta$: $\sin 2\theta = 1 - t^2$

Step 3: Transform the denominator of the integrand. Substitute the expression for $\sin 2\theta$ into the denominator of the integrand: $9 + 16 \sin 2\theta = 9 + 16(1 - t^2)$ $= 9 + 16 - 16t^2$ $= 25 - 16t^2$

Step 4: Change the limits of integration. The original limits are for $\theta$. We need to find the corresponding limits for $t$.

• When $\theta = 0$: $t = \sin 0 - \cos 0 = 0 - 1 = -1$.
• When $\theta = \frac{\pi}{4}$: $t = \sin \frac{\pi}{4} - \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}} = 0$.

Step 5: Rewrite the integral in terms of $t$. Substitute $dt$, the transformed denominator, and the new limits into the integral: $I = 80 \int\limits_{-1}^{0} \frac{dt}{25 - 16t^2}$

Step 6: Prepare the integrand for the standard integration formula. To apply the formula $\int \frac{dx}{a^2 - x^2}$, we need the denominator in the form $a^2 - x^2$. We can factor out 16 from the denominator: $I = 80 \int\limits_{-1}^{0} \frac{dt}{16 \left( \frac{25}{16} - t^2 \right)}$ $I = \frac{80}{16} \int\limits_{-1}^{0} \frac{dt}{\left( \frac{5}{4} \right)^2 - t^2}$ $I = 5 \int\limits_{-1}^{0} \frac{dt}{\left( \frac{5}{4} \right)^2 - t^2}$ Here, $a = \frac{5}{4}$.

Step 7: Apply the standard integration formula. Using the formula $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \ln \left| \frac{a+x}{a-x} \right|$, we get: $I = 5 \left[ \frac{1}{2 \left( \frac{5}{4} \right)} \ln \left| \frac{\frac{5}{4} + t}{\frac{5}{4} - t} \right| \right]_{-1}^{0}$ Simplify the coefficient: $\frac{1}{2 \left( \frac{5}{4} \right)} = \frac{1}{\frac{5}{2}} = \frac{2}{5}$. $I = 5 \left[ \frac{2}{5} \ln \left| \frac{5+4t}{5-4t} \right| \right]_{-1}^{0}$ $I = 2 \left[ \ln \left| \frac{5+4t}{5-4t} \right| \right]_{-1}^{0}$

Step 8: Evaluate the definite integral using the limits. $I = 2 \left( \ln \left| \frac{5+4(0)}{5-4(0)} \right| - \ln \left| \frac{5+4(-1)}{5-4(-1)} \right| \right)$ $I = 2 \left( \ln \left| \frac{5}{5} \right| - \ln \left| \frac{5-4}{5+4} \right| \right)$ $I = 2 \left( \ln |1| - \ln \left| \frac{1}{9} \right| \right)$ Since $\ln 1 = 0$ and $\ln \left( \frac{1}{9} \right) = \ln (9^{-1}) = -\ln 9$: $I = 2 (0 - (-\ln 9))$ $I = 2 \ln 9$ Using the logarithm property $\ln(a^b) = b \ln a$, we can write $\ln 9 = \ln (3^2) = 2 \ln 3$. $I = 2 (2 \ln 3)$ $I = 4 \ln 3$

Common Mistakes & Tips

• Substitution Choice: Ensure the chosen substitution's derivative matches the numerator. If the numerator was $(\cos \theta - \sin \theta)$, the substitution would be $t = \cos \theta + \sin \theta$.
• Changing Limits: Always remember to change the limits of integration when performing a substitution in a definite integral.
• Logarithm Simplification: Be careful with logarithmic properties, especially when evaluating $\ln(1)$ and $\ln(\text{fraction})$. Simplifying $\ln 9$ to $2 \ln 3$ is often necessary to match the options.

Summary

The integral was solved by recognizing a standard trigonometric substitution pattern. By letting $t = \sin \theta - \cos \theta$, the integrand was transformed into a form suitable for the standard integral $\int \frac{dx}{a^2 - x^2}$. After changing the limits of integration and applying the formula, the definite integral was evaluated, yielding $4 \ln 3$.

The final answer is $\boxed{4 \log 3}$.