Question

Evaluate the integral. (Use C for the constant of integra ∫ 1/x 2 36−x 2 ​ ​ dx

Answer 1

The solution of the integral ∫ 1/x 2 36−x 2 ​ ​ dx is 1/6 arcsin(x/6) + C

Step-by-step explanation:

We can evaluate the integral using the trigonometric substitution:

x = 6sinθ

dx = 6cosθ dθ

Substituting these into the integral, we get:

∫ 1/x 2 36−x 2 ​ ​ dx = ∫ 1/(6sinθ) 2 36−(6sinθ) 2 ​ ​ 6cosθ dθ

Simplifying, we get:

∫ 1/(36sin 2θ) 6cosθ dθ

We can now make the substitution:

u = sinθ

du = cosθ dθ

Substituting these into the integral, we get:

∫ 1/(36u 2) 6du

Integrating, we get:

(1/36)u −1 + C

Substituting u = sinθ back, we get:

(1/36)sin −1 θ + C

Finally, substituting x = 6sinθ back, we get:

1/6 arcsin(x/6) + C