How To Integrate Lnx Without Wasting Steps
The correct way to integrate $$\ln(x)$$ is to use integration by parts, with $$u=\ln(x)$$ and $$dv=dx$$, which gives $$\int \ln(x)\,dx = x\ln(x)-x+C$$. That result is the standard antiderivative used in calculus references and step-by-step solvers.
How the method works
Integration by parts follows the identity $$\int u\,dv = uv-\int v\,du$$, and $$\ln(x)$$ is a good choice for $$u$$ because its derivative simplifies to $$1/x$$. In this case, $$dv=dx$$ makes $$v=x$$, so the original integral becomes manageable after substitution.
- Set $$u=\ln(x)$$ and $$dv=dx$$.
- Compute $$du=\frac{1}{x}dx$$ and $$v=x$$.
- Apply $$\int u\,dv = uv-\int v\,du$$.
- Simplify to get $$x\ln(x)-\int 1\,dx$$.
- Finish with $$x\ln(x)-x+C$$.
Worked example
Using the method above, $$\int \ln(x)\,dx = x\ln(x)-\int x\cdot\frac{1}{x}\,dx = x\ln(x)-\int 1\,dx = x\ln(x)-x+C$$. This is the most common form you will see in textbooks, videos, and online integral calculators.
| Step | Expression | Why it helps |
|---|---|---|
| Choose $$u$$ | $$u=\ln(x)$$ | Its derivative becomes simpler. |
| Choose $$dv$$ | $$dv=dx$$ | Its integral is easy. |
| Differentiate/integrate | $$du=\frac{1}{x}dx,\ v=x$$ | Turns the product into a simpler integral. |
| Final answer | $$x\ln(x)-x+C$$ | Standard antiderivative. |
Practical note
If your problem is $$\int \log_b(x)\,dx$$, convert it to natural log first using $$\log_b(x)=\frac{\ln(x)}{\ln(b)}$$, then apply the same result. For definite integrals, evaluate $$x\ln(x)-x$$ at the bounds after finding the antiderivative.
Helpful tips and tricks for How To Integrate Lnx Without Wasting Steps
Why use integration by parts?
Because $$\ln(x)$$ becomes easier to handle after differentiation, while $$dx$$ is easy to integrate. That pairing is exactly why instructors and calculators recommend this method for $$\int \ln(x)\,dx$$.
What is the antiderivative?
The antiderivative of $$\ln(x)$$ is $$x\ln(x)-x+C$$. This is the final expression most calculus resources give for the indefinite integral.
