Key Concepts and Formulas

• Integration of $\frac{f'(x)}{f(x)}$: The integral of a function divided by its derivative is the natural logarithm of the absolute value of the function: $\int \frac{f'(x)}{f(x)} \mathrm{~d} x = \ln|f(x)| + C$.
• Linear Combination for Trigonometric Integrals: Integrals of the form $\int \frac{a \sin x + b \cos x}{c \sin x + d \cos x} \mathrm{~d} x$ can be solved by expressing the numerator as a linear combination of the denominator and its derivative: $a \sin x + b \cos x = A(c \sin x + d \cos x) + B(c \cos x - d \sin x)$.

Step-by-Step Solution

The integral we need to evaluate is: $I = \int_0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x$

Step 1: Express the Numerator as a Linear Combination of the Denominator and its Derivative

Let the numerator be $N(x) = 136 \sin x$ and the denominator be $D(x) = 3 \sin x + 5 \cos x$. We need to find constants $A$ and $B$ such that: $N(x) = A \cdot D(x) + B \cdot D'(x)$ First, let's find the derivative of the denominator: $D'(x) = \frac{\mathrm{d}}{\mathrm{d} x}(3 \sin x + 5 \cos x) = 3 \cos x - 5 \sin x$ Now, we set up the equation: $136 \sin x = A(3 \sin x + 5 \cos x) + B(3 \cos x - 5 \sin x)$ Rearranging the terms on the right-hand side to group $\sin x$ and $\cos x$: $136 \sin x = (3A \sin x - 5B \sin x) + (5A \cos x + 3B \cos x)$ $136 \sin x = (3A - 5B) \sin x + (5A + 3B) \cos x$ By comparing the coefficients of $\sin x$ and $\cos x$ on both sides, we get a system of linear equations:

1. Coefficient of $\sin x$: $3A - 5B = 136 \quad (*)$
2. Coefficient of $\cos x$: $5A + 3B = 0 \quad (**)$

Step 2: Solve the System of Linear Equations for A and B

From equation $(**)$, we can express $A$ in terms of $B$: $5A = -3B \implies A = -\frac{3}{5}B$ Substitute this expression for $A$ into equation $(*)$: $3\left(-\frac{3}{5}B\right) - 5B = 136$ $-\frac{9}{5}B - 5B = 136$ To combine the terms with $B$, find a common denominator: $-\frac{9}{5}B - \frac{25}{5}B = 136$ $\frac{-9B - 25B}{5} = 136$ $\frac{-34B}{5} = 136$ Now, solve for $B$: $-34B = 136 \times 5$ $B = \frac{136 \times 5}{-34}$ Since $136 = 4 \times 34$, we have: $B = \frac{4 \times 34 \times 5}{-34} = -20$ Now substitute the value of $B$ back into the expression for $A$: $A = -\frac{3}{5}B = -\frac{3}{5}(-20) = 3 \times 4 = 12$ So, we have $A = 12$ and $B = -20$.

Step 3: Rewrite the Integral using the Linear Combination

Now we can rewrite the numerator: $136 \sin x = 12(3 \sin x + 5 \cos x) - 20(3 \cos x - 5 \sin x)$ The integral becomes: $I = \int_0^{\pi / 4} \frac{12(3 \sin x + 5 \cos x) - 20(3 \cos x - 5 \sin x)}{3 \sin x+5 \cos x} \mathrm{~d} x$ We can split this into two integrals: $I = \int_0^{\pi / 4} \left( \frac{12(3 \sin x + 5 \cos x)}{3 \sin x+5 \cos x} - \frac{20(3 \cos x - 5 \sin x)}{3 \sin x+5 \cos x} \right) \mathrm{~d} x$ $I = \int_0^{\pi / 4} \left( 12 - 20 \frac{3 \cos x - 5 \sin x}{3 \sin x+5 \cos x} \right) \mathrm{~d} x$

Step 4: Evaluate the Definite Integrals

Now we evaluate each part of the integral separately: $I = \int_0^{\pi / 4} 12 \mathrm{~d} x - 20 \int_0^{\pi / 4} \frac{3 \cos x - 5 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x$

Part 1: $\int_0^{\pi / 4} 12 \mathrm{~d} x$ $\int_0^{\pi / 4} 12 \mathrm{~d} x = [12x]_0^{\pi / 4} = 12\left(\frac{\pi}{4}\right) - 12(0) = 3\pi$

Part 2: $-20 \int_0^{\pi / 4} \frac{3 \cos x - 5 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x$ Let $u = 3 \sin x + 5 \cos x$. Then $\mathrm{d} u = (3 \cos x - 5 \sin x) \mathrm{~d} x$. This integral is of the form $\int \frac{\mathrm{d} u}{u} = \ln|u|$. So, the indefinite integral is: $-20 \int \frac{3 \cos x - 5 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x = -20 \ln|3 \sin x + 5 \cos x|$ Now, we evaluate the definite integral from $0$ to $\pi/4$: $-20 [\ln|3 \sin x + 5 \cos x|]_0^{\pi / 4}$ First, evaluate at the upper limit $x = \pi/4$: $\ln\left|3 \sin\left(\frac{\pi}{4}\right) + 5 \cos\left(\frac{\pi}{4}\right)\right| = \ln\left|3\left(\frac{1}{\sqrt{2}}\right) + 5\left(\frac{1}{\sqrt{2}}\right)\right| = \ln\left|\frac{3+5}{\sqrt{2}}\right| = \ln\left|\frac{8}{\sqrt{2}}\right|$ $\ln\left|\frac{8}{\sqrt{2}}\right| = \ln\left|\frac{8\sqrt{2}}{2}\right| = \ln|4\sqrt{2}|$ Next, evaluate at the lower limit $x = 0$: $\ln|3 \sin(0) + 5 \cos(0)| = \ln|3(0) + 5(1)| = \ln|5| = \ln 5$ Now, subtract the lower limit value from the upper limit value: $-20 \left( \ln|4\sqrt{2}| - \ln 5 \right)$ Using the logarithm property $\ln a - \ln b = \ln(a/b)$: $-20 \ln\left|\frac{4\sqrt{2}}{5}\right|$ Using the logarithm property $k \ln a = \ln(a^k)$: $\ln\left(\left(\frac{4\sqrt{2}}{5}\right)^{-20}\right)$ Alternatively, we can expand $\ln|4\sqrt{2}|$: $\ln|4\sqrt{2}| = \ln(4 \times 2^{1/2}) = \ln(2^2 \times 2^{1/2}) = \ln(2^{2 + 1/2}) = \ln(2^{5/2})$ So, the definite integral part is: $-20 \left( \ln(2^{5/2}) - \ln 5 \right) = -20 \left( \frac{5}{2} \ln 2 - \ln 5 \right)$ $= -20 \times \frac{5}{2} \ln 2 - 20 \times (-\ln 5)$ $= -50 \ln 2 + 20 \ln 5$

Step 5: Combine the Results

Now, add the results from Part 1 and Part 2 to get the total integral $I$: $I = 3\pi + (-50 \ln 2 + 20 \ln 5)$ $I = 3\pi - 50 \ln 2 + 20 \ln 5$ The natural logarithm $\ln$ is equivalent to $\log_e$. $I = 3\pi - 50 \log_e 2 + 20 \log_e 5$

Common Mistakes & Tips

• Sign Errors: Be very careful when solving the system of linear equations for $A$ and $B$, as sign errors can propagate through the entire calculation.
• Logarithm Properties: Incorrectly applying logarithm properties (e.g., $\ln(a+b)$) can lead to wrong answers. Remember $\ln a + \ln b = \ln(ab)$ and $\ln a - \ln b = \ln(a/b)$.
• Derivative of Denominator: Double-check the derivative of the denominator; a mistake here will make the subsequent steps incorrect.

Summary

The integral was evaluated using the standard technique of expressing the numerator as a linear combination of the denominator and its derivative. This decomposed the integral into a constant term and a logarithmic term, which were then evaluated using the fundamental theorem of calculus. The constants $A$ and $B$ were found by solving a system of linear equations derived from matching coefficients of $\sin x$ and $\cos x$. The final result was obtained by combining the evaluated parts of the integral.

The final answer is \boxed{3 \pi-50 \log _e 2+20 \log _e 5}, which corresponds to option (A).