Questions: ∫ 8x cos(5x) dx

∫ 8x cos(5x) dx
Transcript text: \(\int 8 x \cos (5 x) \mathrm{d} x\)
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Solution

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Solution Steps

Step 1: Integration by Parts

To solve the integral \(\int 8 x \cos (5 x) \, dx\), we apply the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] We choose \(u = 8x\) and \(dv = \cos(5x) \, dx\). Then, we compute \(du\) and \(v\): \[ du = 8 \, dx, \quad v = \frac{1}{5} \sin(5x) \]

Step 2: Applying the Formula

Substituting \(u\), \(du\), \(v\), and \(dv\) into the integration by parts formula gives: \[ \int 8 x \cos(5 x) \, dx = 8x \cdot \frac{1}{5} \sin(5x) - \int \frac{1}{5} \sin(5x) \cdot 8 \, dx \] This simplifies to: \[ \frac{8}{5} x \sin(5x) - \frac{8}{5} \int \sin(5x) \, dx \]

Step 3: Integrating the Remaining Integral

The integral \(\int \sin(5x) \, dx\) can be computed as: \[ \int \sin(5x) \, dx = -\frac{1}{5} \cos(5x) \] Thus, substituting this back into our expression gives: \[ \frac{8}{5} x \sin(5x) + \frac{8}{25} \cos(5x) + C \]

Final Answer

The final result of the integral \(\int 8 x \cos(5 x) \, dx\) is: \[ \boxed{\frac{8}{5} x \sin(5x) + \frac{8}{25} \cos(5x) + C} \]

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