Choose Integration by Parts: Let's use integration by parts where u = ln(x) and dv = x^(-3)dx. Differentiate u to get du and integrate dv to get v. du = (1/x)dx and v = -1/(2x^2).Apply Integration by Parts Formula: Now apply the integration by parts formula ∫udv = uv - ∫vdu. Plug in u, du, v into the formula. ∫(1/x^3)ln(x)dx = ln(x)(-1/(2x^2)) - ∫(-1/(2x^2))(1/x)dx.Simplify Integral: Simplify the integral ∫(-1/(2x^2))(1/x)dx. This becomes ∫(-1/(2x^3))dx.Integrate Final Term: Integrate -1/(2x^3) with respect to x. The integral is 1/(4x^2).

Calculus

Find indefinite integrals using the substitution

Full solution

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

Choose Integration by Parts: Let's use integration by parts where $u = \ln(x)$ and $dv = x^{-3}dx$ . Differentiate $u$ to get $du$ and integrate $dv$ to get $v$ . $du = (1/x)dx$ and $v = -1/(2x^2)$ .
Apply Integration by Parts Formula: Now apply the integration by parts formula $\int u\,dv = uv - \int v\,du$ . Plug in $u$ , $du$ , $v$ into the formula. $\int(\frac{1}{x^3})\ln(x)\,dx = \ln(x)(-\frac{1}{2x^2}) - \int(-\frac{1}{2x^2})(\frac{1}{x})\,dx$ .
Simplify Integral: Simplify the integral $\int\left(-\frac{1}{2x^2}\right)\left(\frac{1}{x}\right)dx$ . This becomes $\int\left(-\frac{1}{2x^3}\right)dx$ .
Integrate Final Term: Integrate $-\frac{1}{2x^3}$ with respect to $x$ . The integral is $\frac{1}{4x^2}$ .