Integration by Substitution

Current UK exam textbooks pass over proofs and mathematical discussions in a hurry to show the ‘how to’ of exam questions.

Integration by substitution is a little more than just backwards chain-rule and deserves a fuller treatment.

Try this,

Let, $y=\displaystyle \int \textnormal{f}(x) \ \textnormal{d}x$

then, $\dfrac{\textnormal{d}y}{\textnormal{d}x}=\textnormal{f}(x)$

Suppose that there exists a function g, of another variable $u$, such that $x=\textnormal{g}(u)$ and let, $\textnormal{f}(x)=\textnormal{f}(\textnormal{g}(u))=\textnormal{F}(u)$. So that, $\dfrac{\textnormal{d}y}{\textnormal{d}x}=\textnormal{F}(u)$

Now, by the chain rule, $\dfrac{\textnormal{d}y}{\textnormal{d}u}=\dfrac{\textnormal{d}y}{\textnormal{d}x}\times \dfrac{\textnormal{d}x}{\textnormal{d}u}=\textnormal{F}(u)\dfrac{\textnormal{d}x}{\textnormal{d}u}$

Hence, $y=\displaystyle \int \dfrac{\textnormal{d}y}{\textnormal{d}u} \ \textnormal{d}u=\displaystyle \int \textnormal{F}(u)\dfrac{\textnormal{d}x}{\textnormal{d}u} \ \textnormal{d}u$

i.e. $y=\displaystyle \int \textnormal{f}(x) \ \textnormal{d}x=\displaystyle \int \textnormal{F}(u)\dfrac{\textnormal{d}x}{\textnormal{d}u} \ \textnormal{d}u$

Conic Sections Tangent/Normal Locus

A level further mathematics question animated.  Mid point of Normal x-intercept and Tangent y-intercept traces out another curve as original point moves round an ellipse.

FP3 2014 gives students the chance to prove the surface area of a sphere is 4 pi r squared.  Interestingly four time the area of a cross section through the centre and the derivative of the volume.  Pupils with strong maths general knowledge from GCSE would know what is coming.  TOP TIP do as much maths challenge and extension work as possible, the general knowledge thus gained can only help.  This FP3 2014 paper and many others are at my Exam Bucket: http://jped.co.uk/ExamBucket/exambucket.php.

Useful facts for I/GCSE pupils (exams v soon!) on SIMULTANEOUS EQUATIONS detailed in Carr’s Victorian text book and alive and well, completely unchanged today.  FP1 students look at the determinant in 59.