What is the Appropriate Substitution for Solving the Integral of x/(x^2+2x+2)dx?

In summary, the integral of x/(x2+2x+2)dx can be solved by first completing the square to get x/((x+1)2+1), then trying the substitution u = (x+1) or x+1 = tan u, and finally integrating the resulting fractions separately to get 1/2ln((x+1)2+1) - arctan(x+1) + C.
  • #1
apiwowar
96
0
integral of x/(x2+2x+2)dx

first thing i did was complete the square to get

x/((x+1)2+1

i tried then having x+1 = tanx but that didnt work out

because of the x on top i can't just set w = x+1

what would the right substitution be?

any hints or help would be appreciated
 
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  • #2
apiwowar - try one more substitution, u = (x+1). This will yield two terms in numerator, thus two fractions, for which you should be able to integrate separately with a little more manipulation.
 
  • #3
Using the substitution [itex]x+1=\tan u[/itex] will work just fine. However the substitution in post #2 will make it easier. Show us where you got stuck.
 
  • #4
i did the substitution that the first guy suggested and got 1/2ln((x+1)2+1) - arctan(x+1)

is that right?
 
  • #5
That is what I obtained (don't forget constant of integration).
 

FAQ: What is the Appropriate Substitution for Solving the Integral of x/(x^2+2x+2)dx?

What is integral trig substitution?

Integral trig substitution is a technique used in calculus to evaluate integrals that involve trigonometric functions. It involves substituting a trigonometric expression for a variable in the integrand.

When should I use integral trig substitution?

Integral trig substitution is most useful when the integrand contains a combination of polynomials and trigonometric functions. It can also be used to simplify complex integrals and make them easier to solve.

What are the steps for performing integral trig substitution?

The first step is to identify the appropriate substitution by looking at the integrand and determining which trigonometric function would be most useful. Then, make the substitution and simplify the integral. Next, use trigonometric identities to rewrite the integral in terms of the new variable. Finally, solve the integral and substitute back in the original variable to get the final answer.

What are some common trigonometric identities used in integral trig substitution?

Some common trigonometric identities used in integral trig substitution include the Pythagorean identities, double-angle identities, and half-angle identities. These identities can help simplify the integral and make it easier to solve.

Are there any special cases I should be aware of when using integral trig substitution?

Yes, there are a few special cases to be aware of when using integral trig substitution. For example, if the integrand contains both sine and cosine functions, it may be helpful to use a tangent substitution. Also, if the integrand contains a square root term, you may need to use a trigonometric substitution in addition to integral trig substitution.

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