Trigonometric Substitution...4

May 2016
368
5
NYC
Integrate -5x/(x^2+5)^(3/2) dx

Let S = integral symbol

-5 S x/[root(x^2+5)]^3 dx

x = root(5)tanθ

dx = root(5)sec^2 θ dθ

Can I get the next three steps?
I can take it from there. Thanks.
 

Prove It

MHF Helper
Aug 2008
12,897
5,001
You should always look for SIMPLE substitutions before trying a more difficult trigonometric/hyperbolic substitution.

$\displaystyle \begin{align*} \int{ -\frac{5\,x}{\left( x^2 + 5 \right) ^{\frac{3}{2}}}\,\mathrm{d}x} &= -\frac{5}{2} \int{ 2\,x\,\left( x^2 + 5 \right) ^{-\frac{3}{2}}\,\mathrm{d}x } \end{align*}$

Let $\displaystyle \begin{align*} u = x^2 + 5 \implies \mathrm{d}u = 2\,x\,\mathrm{d}x \end{align*}$ and the integral becomes

$\displaystyle \begin{align*} -\frac{5}{2}\int{2\,x\,\left( x^2 + 5 \right) ^{-\frac{3}{2}}\,\mathrm{d}x} &= -\frac{5}{2}\int{ u^{-\frac{3}{2}}\,\mathrm{d}u} \end{align*}$

Go from here.
 
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May 2016
368
5
NYC
Much easier now. I can take it from here.