How do you integrate #int dx/sqrt(x^2+4)# using trig substitutions?

Question

Evelyn Agard · Accepted Answer

Substitute #x=2 sinh u# quickly getting the integral to be #sinh^{-1}(x/2)+C# #x=2sinh u#, #dx=2cosh u du# giving the integral as #int2coshu/ sqrt(4 sinh^2u+4)du# #=int cosh u/ cosh u du# because #cosh^2u -sinh^2 u =1# #=int1du# #=u+C# #=sinh^-1(x/2)+C# Some people prefer to write this as #ln(x+sqrt(x^2+4))# which is equivalent, as it differs from the first solution by a fixed constant #ln(1/2)#.

Elijah Abram · Answer

undefined To integrate ∫ dx / sqrt(x^2 + 4) using trigonometric substitution, let x = 2tan(θ), then dx = 2sec^2(θ) dθ. Substitute these into the integral:

∫ (2sec^2(θ) dθ) / sqrt((2tan(θ))^2 + 4)

Simplify the expression under the square root:

∫ (2sec^2(θ) dθ) / sqrt(4tan^2(θ) + 4)
= ∫ (2sec^2(θ) dθ) / sqrt(4(sec^2(θ)) + 4)
= ∫ (2sec^2(θ) dθ) / sqrt(4sec^2(θ) + 4)

Combine like terms under the square root:

= ∫ (2sec^2(θ) dθ) / sqrt(4(sec^2(θ) + 1))

Now, using the identity sec^2(θ) + 1 = tan^2(θ) + 1 = sec^2(θ), we simplify further:

= ∫ (2sec^2(θ) dθ) / (2sec(θ))
= ∫ sec(θ) dθ

The integral of sec(θ) with respect to θ is ln|sec(θ) + tan(θ)| + C.

Finally, substitute back for θ using the original substitution x = 2tan(θ):

ln|sec(θ) + tan(θ)| + C
= ln|sec(arctan(x/2)) + tan(arctan(x/2))| + C
= ln|√(x^2 + 4) + x/2| + C

Evelyn Agard · Answer

undefined To integrate $ \int \frac{dx}{\sqrt{x^2 + 4}} $ using trigonometric substitution, we make the substitution $ x = 2\tan(\theta) $.

Then, $ dx = 2\sec^2(\theta) \, d\theta $, and $ \sqrt{x^2 + 4} = \sqrt{4\tan^2(\theta) + 4} = 2\sec(\theta) $.

Substituting these into the integral, we get:

$$ \int \frac{2\sec^2(\theta) \, d\theta}{2\sec(\theta)} $$

$$ = \int \sec(\theta) \, d\theta $$

The integral of $ \sec(\theta) $ can be evaluated using the formula $ \int \sec(\theta) \, d\theta = \ln|\sec(\theta) + \tan(\theta)| + C $.

Thus, the integral $ \int \frac{dx}{\sqrt{x^2+4}} $ can be expressed as $ \ln|\sec(\theta) + \tan(\theta)| + C $, where $ C $ is the constant of integration.