Questions: ∫ (cos(5x)) / (4sin(5x) + 7) dx =

∫ (cos(5x)) / (4sin(5x) + 7) dx =
Transcript text: \[ \int \frac{\cos (5 x)}{4 \sin (5 x)+7} d x= \]
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Solution

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Solution Steps

To evaluate the integral \(\int \frac{\cos (5 x)}{4 \sin (5 x)+7} \, dx\), we can use a substitution method. Let \(u = 4 \sin(5x) + 7\). Then, find \(du\) and rewrite the integral in terms of \(u\).

Step 1: Substitution

Let \( u = 4 \sin(5x) + 7 \). Then, we need to find \( du \).

Step 2: Differentiate \( u \)

Differentiate \( u \) with respect to \( x \): \[ \frac{du}{dx} = 4 \cdot 5 \cos(5x) = 20 \cos(5x) \] Thus, \[ du = 20 \cos(5x) \, dx \]

Step 3: Rewrite the Integral

Rewrite the integral in terms of \( u \): \[ \int \frac{\cos(5x)}{4 \sin(5x) + 7} \, dx = \int \frac{1}{20} \cdot \frac{1}{u} \, du \]

Step 4: Integrate

Integrate with respect to \( u \): \[ \int \frac{1}{20} \cdot \frac{1}{u} \, du = \frac{1}{20} \int \frac{1}{u} \, du = \frac{1}{20} \ln|u| + C \]

Step 5: Substitute Back

Substitute \( u = 4 \sin(5x) + 7 \) back into the integral: \[ \frac{1}{20} \ln|4 \sin(5x) + 7| + C \]

Final Answer

\[ \boxed{\frac{1}{20} \ln|4 \sin(5x) + 7| + C} \]

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