s- integral
s 1/ [ (x^4) sq.rt(x^2+9)]
i know x=3tanx
sq.rt(x^2+9)= 3 secx
dx= 3/[cos^2(x)]
so far i know:
= 1/ (3tan^4(x)) 3secx cos^2(x)) dx
=1/ 81 [ (sin^4 (x)/cos^4 (x)) (1/cosx) (cos^2(x))]
then i'm not really sure what to do next
Well, it seems like you have made some progress! Let's take a look at the integral you have:
∫ [1 / (x^4 √(x^2 + 9))] dx
Using the trigonometric substitution x = 3tanθ, we have:
x^2 + 9 = (3tanθ)^2 + 9 = 9(tan^2θ + 1) = 9sec^2θ
Now, let's rewrite the integral in terms of θ:
∫ [1 / ((3tanθ)^4 √(9sec^2θ))] (3sec^2θ) dθ
Simplifying this further:
∫ [(1 / 81) * (tan^4θ / sec^4θ) * sec^2θ] dθ
= (1 / 81) ∫ [tan^4θ] dθ
Now, you can integrate tan^4θ by using the power-reducing formula for tan:
tan^2θ = sec^2θ - 1
Multiply both sides by tan^2θ:
tan^4θ = (sec^2θ - 1)^2
= sec^4θ - 2sec^2θ + 1
Now, substitute this back into the integral:
(1 / 81) ∫ [(sec^4θ - 2sec^2θ + 1)] dθ
= (1 / 81) ∫ [sec^4θ] dθ - (2 / 81) ∫ [sec^2θ] dθ + (1 / 81) ∫ dθ
Now, integrate each term separately. The integral of sec^2θ is tanθ, and the integral of dθ is just θ:
= (1 / 81) ∫ [sec^4θ] dθ - (2 / 81) ∫ [sec^2θ] dθ + (1 / 81) θ + C
Congratulations on making it this far! Now, try completing the integrals for sec^4θ and sec^2θ, and then you can simplify the expression further if needed. Good luck!
To continue solving the integral, you can simplify the expression further and then use the appropriate trigonometric identities.
Let's go through the steps:
1. Start with the expression:
∫ 1/[x^4 √(x^2+9)] dx
2. Substitute x = 3tanθ, which gives:
√(x^2+9) = 3secθ
To find dx in terms of θ, differentiate both sides of the substitution equation:
dx = 3sec^2θ dθ
3. Substitute back into the integral:
∫ 1/[x^4 √(x^2+9)] dx = ∫ [1/(3tanθ)^4 * 3secθ * 3sec^2θ] dθ
4. Simplify the expression:
∫ [1/(27tan^4θsecθ)] dθ = ∫ [sec^-1θ/(27tan^4θ)] dθ
5. Apply the trigonometric identity: sec^2θ = 1 + tan^2θ
sec^-1θ = 1/√(1 + tan^2θ), and substitute into the integral:
∫ [ (1/√(1 + tan^2θ)) / (27tan^4θ)] dθ
6. Apply the power reducing formula for tangent: tan^2θ = sec^2θ - 1
Substitute into the integral:
∫ [ (1/√(1 + (sec^2θ - 1))) / (27tan^4θ)] dθ
∫ [ (1/√(sec^2θ)) / (27tan^4θ)] dθ
∫ [ (1/secθ) / (27tan^4θ)] dθ
7. Simplify further:
∫ [ (cosθ/ (27sin^4θcos^3θ))] dθ
∫ [ (1/ (27sin^4θcos^2θ))] dθ
Now you have an integral in terms of θ that you can solve by applying basic integration techniques.