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You must avoid these common integration errors in your Maths Extension 2 exams.
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Avoid the 6 Common Errors In Year 12 Maths Extension 2 Integration! Learn the errors HSC students are making and avoid them.
Confusion with the difference between \((ln \ x){^2}\) and \( ln \ x{^2}\) and consequently errors made in differentiation or integration of these functions.
These two functions are not equal!
\( ln \ x{^2} = 2 \ ln \ x\) whereas \((ln \ x){^2} = (ln \ x) \times (ln \ x)\)
Hence, \( \frac {d}{dx} (ln \ x{^2}) = \frac{d}{dx} [2 \ ln \ x] = 2 \times \frac{1}{x}\)
\(\frac{d}{dx}(ln \ x{^2}\) requires the use of the chain rule.
\(\frac{d}{dx} (ln \ x){^2} = 2 \ ln \ x \times \frac{1}{x}\)
Incorrect choice of the function for \( \frac{dv}{dx}\) and for \(u\).
LIATE (Logarithmic, Inverse, Trigonometric, Exponential) is an order that you could use to determine which function is represented by \(u\) when the integrand is a product of any two of these functions.
\( \int x{^2} dx\) use \(u= ln \ x\)\(\frac{dv}{dx}=x{^2}\) (since \(ln \ x\) and for \(x{^2}\) is algebraic)
Other common errors are from \(\frac{dv}{dx}\) you need to integrate and not differentiate to get \(v\) and you need to differentiate \(u\) to ger \(\frac{du}{dx}\).
\(\frac{dv}{dx}=x{^2} \) therefore \(v=2x\) (you have differentiated, not integrated). It should be \(\frac{dv}{dx}=x{^2}\) then \(v=\frac{x{^3}}{3}\)
In your solutions for the integral of products involving the use of integration by parts:
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When integrating products, first check that one of the functions cannot be expressed as a derivative of the other function. If it can, then integration by parts is not the correct technique. An appropriate substitution may lead to a standard integral.
\(\int e{^{\cos x}}\sin x \ dx\) By using the substitution \(u= \cos x\) will lead to a standard integral.
When using a substition in indefinite integrals, the final answer is often not rewritten in terms of the original variable.
\(\int\frac{dx}{1+\cos x}\) using the substition \(t=\tan\frac{x}{2}\). The substitution changes the variable back to \(x\) using \(t=\tan\frac{x}{2}\)
For definite integrals, one of the common errors is the failure to also change the limits. There is no need to rewrite the answer in terms of the original variable for definite integrals.
Evaluate \(\int_1^3 \frac{dx}{x\sqrt{2x+1}}\)
Using the substitution \(u{^2}=2x+1\), the limits will change from \(x=1\) to \(u=\sqrt{3}\) and \(x=3\) to \(u=\sqrt{7}\).
These tips were written by Matrix Education’s former Head Maths Teacher Pat Rockett.
Need help with Maths Ext 2?
Expert teachers, weekly quizzes, one-to-one help! Ace your next Maths Ext 2 assessment with Matrix+ Online.
Written by Oak Ukrit
Oak is the Head of Mathematics at Matrix Education and has been teaching for over 12 years and has been helping students at Matrix since 2016. He has 1st class honours in Aeronautical Engineering from UNSW where he taught for over 4 years while he was undertaking a PhD. When not plane spotting he enjoys landscape photography.© Matrix Education and www.matrix.edu.au, 2023. Unauthorised use and/or duplication of this material without express and written permission from this site’s author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Matrix Education and www.matrix.edu.au with appropriate and specific direction to the original content.