Many students often overlook that the coefficient of x in the integration or differentiation formulas in MF15 is 1!!! When it is not 1, many things changes. I’ll let the examples do the talking. 🙂

Differentiation (recall chain rule)

\frac {d}{dx}(sin^{-1}(3x^2)) = \frac {1}{\sqrt{1-(3x^2)^2}}(6x)

Integration

\int \frac {1}{4+9x^2} dx = \frac {1}{2} {tan^{-1}(\frac {3x}{2})}\times\frac{1}{3}

For my careless students, I usually recommend they make the case of the coefficient of x be ONE instead. So \int \frac {1}{4+9x^2} dx = \frac{1}{9}\int \frac {1}{\frac{4}{9}+x^2} dx and after applying formula gives, (\frac{1}{9})(\frac{1}{\frac{2}{3}}){tan^{-1}(\frac {x}{\frac{2}{3}})} which will give the same answers after simplifications.

Practice

You may practice a few of the following questions!

\int\frac {1}{4-9x^2}dx

\int\frac {1}{9x^{2}-4}dx

\int\frac {1}{\sqrt{4-9x^2}}dx

\int\frac {1}{2x^{2}-2x-10}dx

Let me know if you have problems!

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