# Read This Simple Work for Definite Integrals by U-SUBSTITUTION Method Explained in 3 Distinct Examples and Video Lessons

U-substitution in definite integrals is very similar to the method in indefinite integrals, while the initial bounds (usually x-values) need to be changed to the corresponding u-values for upper and lower limits. You need to account for the limits of the integration. Alternatively, you can integrate the integral expressions using u-substitutions, then change u-expressions back […]

# Definite Integral of Rational Functions

\begin{align} \displaystyle \int_{n}^{m}{\dfrac{1}{x}}dx &= \big[\log_e{x}\big]_{n}^{m} \\ &= \log_{e}{m} – \log_{e}{n} \end{align} Generally, \begin{align} \displaystyle \int_{n}^{m}{\dfrac{f'(x)}{f(x)}}dx &= \big[\log_e{f(x)}\big]_{n}^{m} \\ &= \log_{e}{f(m)} – \log_{e}{f(n)} \end{align} Example 1 Find $\displaystyle \int_{1}^{5}{\dfrac{2}{x}}dx$. \( \begin{align} \displaystyle \int_{1}^{5}{\dfrac{2}{x}}dx &= 2\int_{1}^{5}{\dfrac{1}{x}}dx \\ &= 2\big[\log_{e}{x}\big]_{1}^{5} \\ &= 2 \log_{e}{5} – 2 \log_{e}{1} \\ &= 2 \log_{e}{5} – 2 \times […]