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int(1)/(x^(3))ln xdx

∫1x3ln⁡x dx\int \frac{1}{x^{3}}\ln x \, dx

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Full solution

Q. ∫1x3ln⁡x dx\int \frac{1}{x^{3}}\ln x \, dx
  1. Choose Integration by Parts: Let's use integration by parts where u=ln⁡(x)u = \ln(x) and dv=x−3dxdv = x^{-3}dx. Differentiate uu to get dudu and integrate dvdv to get vv. du=(1/x)dxdu = (1/x)dx and v=−1/(2x2)v = -1/(2x^2).
  2. Apply Integration by Parts Formula: Now apply the integration by parts formula ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du. Plug in uu, dudu, vv into the formula. ∫(1x3)ln⁡(x) dx=ln⁡(x)(−12x2)−∫(−12x2)(1x) dx\int(\frac{1}{x^3})\ln(x)\,dx = \ln(x)(-\frac{1}{2x^2}) - \int(-\frac{1}{2x^2})(\frac{1}{x})\,dx.
  3. Simplify Integral: Simplify the integral ∫(−12x2)(1x)dx\int\left(-\frac{1}{2x^2}\right)\left(\frac{1}{x}\right)dx. This becomes ∫(−12x3)dx\int\left(-\frac{1}{2x^3}\right)dx.
  4. Integrate Final Term: Integrate −12x3-\frac{1}{2x^3} with respect to xx. The integral is 14x2\frac{1}{4x^2}.

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