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Mathematics

Integrating Products with Parts

Mastering the integration by parts technique to solve the integral of x sin(5x) by breaking a product into manageable pieces.

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The Challenge of Products
The Product Trapf(x) * g(x)
∫xsin⁡(5x) dx\int x \sin(5x) \, dx
Have you ever tried to integrate a product like x sin(5x)? Standard rules don't quite reach here, because the x and the sine term are locked together.
Step-by-step solver
1

Identify components

Set u = x to simplify via differentiation and dv = sin(5x)dx to integrate.

u=x,dv=sin⁡(5x)dx  ⟹  du=dx,v=−15cos⁡(5x)u = x, dv = \sin(5x)dx \implies du = dx, v = -\frac{1}{5}\cos(5x)
2

Apply the Parts Formula

Substitute into the integration by parts identity.

∫0π/2xsin⁡(5x)dx=[−x5cos⁡(5x)]0π/2−∫0π/2−15cos⁡(5x)dx\int_0^{\pi/2} x\sin(5x) dx = [-\frac{x}{5}\cos(5x)]_0^{\pi/2} - \int_0^{\pi/2} -\frac{1}{5}\cos(5x) dx
3

Simplify and evaluate

Calculate the boundary values and solve the remaining integral.

−π/25cos⁡(5π2)−(0)+15∫0π/2cos⁡(5x)dx=0+15[15sin⁡(5x)]0π/2=125sin⁡(5π2)=125-\frac{\pi/2}{5}\cos(\frac{5\pi}{2}) - (0) + \frac{1}{5} \int_0^{\pi/2} \cos(5x) dx = 0 + \frac{1}{5} [\frac{1}{5}\sin(5x)]_0^{\pi/2} = \frac{1}{25}\sin(\frac{5\pi}{2}) = \frac{1}{25}

Original question

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