Resources for Application Of Integration 1 over x
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Application Of Integration 1 over x Theory
![Applications of integration of reciprocal functions involves determining areas involving the \(x\) and \(y\) axes.\\ \begin{multicols}{2} \textbf{Example 1}\\ Find the exact area bounded by the curve \(y=\dfrac{1}{x+1}\), the \(x\)-axis and the lines \(x=7\) and \(x=1\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} \text { Area } & =\displaystyle \int_1^7 \frac{1}{x+1} d x \\ & =\left[\log _e|x+1|\right]_1^7 \\ & =\log _e 8-\log _e 2 \\ & =\log _e 4 \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ Find the exact area bounded by the curve \(y=\dfrac{1}{x-1}\), the \(y\)-axis and the lines \(y=1\) and \(y=\dfrac{1}{2}\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} & y=\frac{1}{x-1} \\ & x-1=\frac{1}{y} \\ & x=\frac{1}{y}+1 \end{aligned}$\\ $\begin{aligned} \text { Area } & =\displaystyle \int_{\frac{1}{2}}^1 \frac{1}{y}+1 \,d y \\ & =\left[\log _e y+y\right]_{\frac{1}{2}}^1 \\ & =\left[\log _e 1+1-\left(\log _e \frac{1}{2}+\frac{1}{2}\right)\right] \\ & =1-\log _e \frac{1}{2}-\frac{1}{2} \\ & =\frac{1}{2}+\log _e 2 \end{aligned}$\\ \end{multicols}](/media/wraf0nft/4787.png)