# Math Help - How do you spot it's a substitution

1. ## How do you spot it's a substitution

$\int^{9}_{5} \frac{x}{\sqrt{(x-5)^{2}+4^{2}}}$

Now, according to my mark scheme this is not by parts, but using the substitution u = x-5. How on earth was I ment to spot that?? How would you maths gurus spot something like that

Thanks

2. Originally Posted by thomas49th
$\int^{9}_{5} \frac{x}{\sqrt{(x-5)^{2}+4^{2}}}$

Now, according to my mark scheme this is not by parts, but using the substitution u = x-5. How on earth was I ment to spot that?? How would you maths gurus spot something like that

Thanks
$\int_{5}^{9} \frac{x}{\sqrt{(x-5)^{2}+4^{2}}}$

u=x-5 du=dx

$\int_{0}^{4} \frac{u+5}{u^2+4^{2} } du$

let $2tan(y) = u \Rightarrow 2sec^2y dy =du$

$2tan(y)=u \Rightarrow 4tan^2 y =u^2 \Rightarrow 4(sec^2y -1 ) =u^2 \Rightarrow sec^2y = u^2+4$

you will have

$\int \left(\frac{2tan(y) +5 }{sec^2 y }\right)\left(2sec^2y dy \right) dy$

can you continue