Why Is x/5 the Right Substitution for Integrating 1/(25+x^2)?

[Swerting]

Homework Statement

I was asked to find the formula for the antiderivative $$\int1/(25+x^{2})$$

Homework Equations

Take a 'part' of the equation and use it to solve the antiderivative, integration by substitution.
and $$dw=(1/5)dx$$

The Attempt at a Solution

I initially set my substitution variable, w, to $$w=25+x^{2}$$, but this did not work out when I tried to differentiate and solve. When I looked at the hint button, it suggested setting w to $$w=x/5$$. I didn't understand how this would help, and when I finally gave in and looked at the solution, I was even more confused. I don't understand how using $$w=x/5$$ is the appropriate substitution.

 

[rock.freak667]

try a trig substitution of x=Atanθ knowing that 1+tan²θ=sec²θ (the value of A should be clear now)

 

[Swerting]

I'm beginning to understand it, I just need to mull it over for a couple minutes. Thank you very much for your help!

 

[Mark44]

The hint you are given assumes that you know this antiderivative formula:
$$\int \frac{du}{u^2 + 1}~=~tan^{-1}(u) + C$$

rock.freak's suggestion can be used if you don't know the formula above.

 

[darkmagic]

I think its 1/a arctan u +c. a=1, so its not written there.

 

[Elucidus]

Homework Statement

I was asked to find the formula for the antiderivative $$\int1/(25+x^{2})$$

Homework Equations

Take a 'part' of the equation and use it to solve the antiderivative, integration by substitution.
and $$dw=(1/5)dx$$

The Attempt at a Solution

I initially set my substitution variable, w, to $$w=25+x^{2}$$, but this did not work out when I tried to differentiate and solve. When I looked at the hint button, it suggested setting w to $$w=x/5$$. I didn't understand how this would help, and when I finally gave in and looked at the solution, I was even more confused. I don't understand how using $$w=x/5$$ is the appropriate substitution.

Determining which integration method to use is often tricky and a process for which there does not exist a recipe. Once a substitution method is chosen, figuring out what expression to substitute for is often non-obvious. The integral above is usually evaluated using a trigonometric reverse substitution. However the hint to use x/5 works (was it a hint?).

Even after the substitution u = x/5 is carried out, there is the assumption that 1/(1 + u²) has a recognizable antiderivative.

Many of my students are frustrated by what appears to them to be arcane guidelines and special cases for integrating. Unfortunately integration is an often difficult process and an antiderivative is frequently elusive or worse. For example, there is no closed form for $\int \ln(x)/x \; dx$.

Skill at integrating (including selecting wise substitutions) is acquired through experience and instructional guidance. The exercise you've presented is a nice example of a case involving a non-obvious substitution.

I leave you with the example:

$$\text{Evaluate }\int \sec(x) \; dx \text{ using the substitution }u = \sec(x) + \tan(x)$$

Very obvious, yes?

--Elucidus