How do you use trig substitution to evaluate integral #x/sqrt(x^2+x+1)dx#?

Question

Isabella Ackerman · Accepted Answer

#int x/sqrt(x^2+x+1) dx = sqrt(x^2+x+1) -1/2 ln abs (2x+1+ 2sqrt( x^2+x+1) )+C#

Write the numerator of integrand as:

#x = 1/2(2x+1) -1/2#

and note that:

#d/dx (x^2+x+1) = 2x+1#

Then:

#int x/sqrt(x^2+x+1) dx = 1/2 int (2x+1)/sqrt(x^2+x+1)dx -1/2 int dx/sqrt(x^2+x+1)#

#int x/sqrt(x^2+x+1) dx = 1/2 int (d(x^2+x+1))/sqrt(x^2+x+1)dx -1/2 int dx/sqrt(x^2+x+1)#

#int x/sqrt(x^2+x+1) dx = sqrt(x^2+x+1) -1/2 int dx/sqrt(x^2+x+1)#

Solve now the resulting integral by completing the square at the denominator:

#1/2 int dx/sqrt(x^2+x+1) = 1/2 int dx/sqrt( (x+1/2)^2 +3/4)#

#1/2 int dx/sqrt(x^2+x+1) = int dx/sqrt( (2x+1)^2 +3)#

Substitute now:

#2x+1 = sqrt3 tant# with #t in (-pi/2,pi/2)#

#dx = sqrt3/2 sec^2 t dt#

to have:

#1/2 int dx/sqrt(x^2+x+1) = sqrt3/2 int (sec^2 t dt)/sqrt( 3tan^2t+3)#

#1/2 int dx/sqrt(x^2+x+1) = 1/2 int (sec^2 t dt)/sqrt( tan^2t+1)#

Use the trigonometric identity:

#tan^2t +1 = sec^2t#

and as for #t in (-pi/2,pi/2)# the secant is positive:

#sqrt(tan^2t +1) = sect#

Then:

#1/2 int dx/sqrt(x^2+x+1) = 1/2 int (sec^2 t dt)/sect#

#1/2 int dx/sqrt(x^2+x+1) = 1/2 int sec t dt#

#1/2 int dx/sqrt(x^2+x+1) = 1/2 ln abs (sect+tant) +C#

and undoing the substitution:

#1/2 int dx/sqrt(x^2+x+1) = 1/2 ln abs ((2x+1)/sqrt3 + sqrt( ( 2x+1)^2/3+1)) +C#

#1/2 int dx/sqrt(x^2+x+1) = 1/2 ln abs ((2x+1)+ sqrt( ( 2x+1)^2+3)) +C#

#1/2 int dx/sqrt(x^2+x+1) = 1/2 ln abs (2x+1+ 2sqrt( x^2+x+1) )+C#

Putting the partial solutions together:

#int x/sqrt(x^2+x+1) dx = sqrt(x^2+x+1) -1/2 ln abs (2x+1+ 2sqrt( x^2+x+1) )+C#

Christopher Allers · Answer

undefined To evaluate the integral $\int \frac{x}{\sqrt{x^2 + x + 1}} dx$ using trigonometric substitution, we can let $x = \tan^2(\theta)$. This substitution can simplify the integral by expressing it in terms of trigonometric functions.

By making this substitution, we can rewrite the integral in terms of $\theta$ and then integrate with respect to $\theta$. After finding the antiderivative, we can then substitute back in terms of $x$.

Here are the steps:

1. Start by making the substitution $x = \tan^2(\theta)$.

2. Calculate $dx$ in terms of $d\theta$ using the derivative of $\tan^2(\theta)$.

3. Substitute $x$ and $dx$ in terms of $\theta$ into the original integral.

4. Simplify the expression inside the integral.

5. Integrate with respect to $\theta$.

6. Substitute back in terms of $x$ using the original substitution $x = \tan^2(\theta)$.

7. Simplify the expression if necessary.

8. Evaluate the integral over the given limits of integration.

9. Finally, solve for the value of the integral.