Calculate the definite integral $$\displaystyle \int_{1}^e \frac{(\ln(x))^3}{x} \ dx$$ , in the interval $$[1, e]$$.
See development and solution
Development:
We will proceed the following way:
- Find the primitive function of the integrand.
$$\displaystyle \int (\ln(x))^3\cdot\dfrac{1}{x} \ dx= \dfrac{(\ln(x))^4}{4}$$
- Evaluate it at the ends of the interval of integration.
$$\Big[\dfrac{(\ln(x))^4}{4}\Big]^e_1=\dfrac{(\ln(e))^4}{4}-\dfrac{(\ln(1))^4}{4}=\dfrac{1}{4}-0=\dfrac{1}{4}$$
Solution:
$$\displaystyle \int_{1}^e \frac{(\ln(x))^3}{x} \ dx=\dfrac{1}{4}$$