Question

Multiple Choice \(\int x \sin (5 x) d x=\) (A) \(-x \cos (5 x)+\sin (5 x)+C\) (B) \(-\frac{x}{5} \cos (5 x)+\frac{1}{25} \sin (5 x)+C\) (C) \(-\frac{x}{5} \cos (5 x)+\frac{1}{5} \sin (5 x)+C\) (D) \(\frac{x}{5} \cos (5 x)+\frac{1}{25} \sin (5 x)+C\) (E) \(5 x \cos (5 x)-\sin (5 x)+C\)

Short answer

Therefore, the integral \(\int x \sin (5 x) d x = -\frac{1}{5}x \cos(5x) + \frac{1}{25}\sin(5x) + C \), which corresponds to option (B).

Step-by-step solution

Identify 'u' and 'dv'

The first step in using the integration by parts formula is to identify 'u' and 'dv'. Here, let's take \(u = x\) and \(dv = \sin(5x) dx\) . This choice is made as the derivative of 'x' is simpler, i.e., '1' and easy to integrate \(\sin(5x)\).

Compute 'du' and 'v'

Having identified 'u' and 'dv', we now calculate 'du' and 'v'. For \(u = x\), \(du = dx\), and \(v = \int dv = -\frac{1}{5} \cos(5x)\).

Apply Integration by Parts

Now we apply the integration by parts formula: \(\int u dv = uv - \int v du\). This results in \(\int x \sin(5x) dx = x (-\frac{1}{5}\cos(5x)) - \int (-\frac{1}{5}\cos(5x)) dx\).

Simplify and Calculate the Remaining Integral

Simplifying gives \(-\frac{1}{5}x \cos(5x) + \frac{1}{5}\int \cos(5x) dx\). The integral of \(\cos(5x)\) can be calculated directly which results in \(\frac{1}{5} \sin(5x)\).

Combine Outputs

So, our final answer is \(-\frac{1}{5}x \cos(5x) + \frac{1}{25}\sin(5x) + C\) (Remember to add '+ C' to signify the constant of integration.)