Resources for Integration of Reciprocal Function
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Questions
15
With Worked SolutionClick Here -
Video Tutorials
1
Click Here -
HSC Questions
2
With Worked SolutionClick Here
Integration of Reciprocal Function Theory
![\(\displaystyle \int \frac{f^{\prime}(x)}{f(x)} d x=\log _e f(x)+c\)\\ \begin{multicols}{2} \textbf{Example 1}\\ %6 (30975) Find the primitive of \(\displaystyle \frac{x^2}{x^3 + 1}\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} \int \, \frac{x^2}{x^3 + 1} \, dx &= \frac{1}{3} \int \, \frac{3x^2}{x^3 + 1} \, dx\\&= \frac{1}{3}\ln(x^3 + 1) + C \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ %24756 \begin{itemize} \item[\bf{i)}]Differentiate \(x\ln x\) \item[\bf{ii)}]Hence evaluate \(\displaystyle \int\limits_1^2 {\ln x\,\,dx} \) \end{itemize} \textbf{Example 2 solution}\\ $\begin{aligned} &\begin{aligned}\text { (i) }\quad & y=x \ln x \\ & u=x \quad v=\ln x \\ & d u=1 \quad d v=\frac{1}{x} \\ & y^{\prime}=1+\ln x \end{aligned}\\ &\begin{aligned}\text { (ii) }\quad \int_{1}^{2} 1+\ln x d x &=[x \ln x]_{1}^{2} \\ \int_{1}^{2} 1 d x+\int_{1}^{2} \ln x d x &=[x \ln x]_{1}^{2} \\ \therefore \int_{1}^{2} \ln x d x &=[x \ln x-x]_{1}^{2} \\ &=[(2 \ln 2-2)-(0-1)] \\ &=\ln 4-1 \end{aligned} \end{aligned}$\\ \end{multicols}](/media/i0mjlfdq/4786.png)