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Evaluate integral { \int \frac{logz}{z}dz }

Question:

Evaluate integral

{eq}\int \frac{logz}{z}dz {/eq}

Integral

This question is from the indefinite integration and we have to integrate this and add constant C after integrating this. This question will be solved by using the substitution method.

Answer and Explanation:

{eq}\Rightarrow \ I=\int\frac{log{z}}{z}dz\\ \text{let,} \Rightarrow \ log{z}=t\\ \text{differentiate with resoect to z }\\ \Rightarrow \ \frac{1}{z}=\frac{dt}{dz}\\ \Rightarrow \ dz=zdt\\ \Rightarrow \ I=\int{t}dt\\ \Rightarrow \ I=[\frac{t^{2}}{2} \ ]+C\\ \text{undo substitution}\\ \Rightarrow \ I=\frac{(log^{2}{z})}{2}+C\\ {/eq}


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