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intx^(-1//2)sin xdx

∫x−1/2sin⁡xdx \int x^{-1 / 2} \sin x d x

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Full solution

Q. ∫x−1/2sin⁡xdx \int x^{-1 / 2} \sin x d x
  1. Identify integral: Identify the integral to be solved.\newline∫x−12sin⁡(x) dx\int x^{-\frac{1}{2}} \sin(x) \, dx
  2. Use integration by parts: Use integration by parts, where u=x−12u = x^{-\frac{1}{2}} and dv=sin⁡(x) dxdv = \sin(x) \, dx.du=−12x−32 dxdu = -\frac{1}{2} x^{-\frac{3}{2}} \, dx and v=−cos⁡(x)v = -\cos(x)
  3. Apply integration by parts formula: Apply the integration by parts formula: ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du. ∫x(−1/2)sin⁡(x) dx=−x(−1/2)cos⁡(x)−∫−cos⁡(x)⋅(−1/2x(−3/2)) dx\int x^{(-1/2)} \sin(x) \, dx = -x^{(-1/2)} \cos(x) - \int -\cos(x) \cdot (-1/2 x^{(-3/2)}) \, dx
  4. Simplify integral: Simplify the integral.\newline=−x−12cos⁡(x)+12∫x−32cos⁡(x) dx= -x^{-\frac{1}{2}} \cos(x) + \frac{1}{2} \int x^{-\frac{3}{2}} \cos(x) \, dx
  5. Attempt to solve remaining integral: Attempt to solve the remaining integral ∫x(−3/2)cos⁡(x) dx\int x^{(-3/2)} \cos(x) \, dx. This integral is complex and typically requires special functions or numerical methods for exact solutions.

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