Question

Evaluate the integral: z=x³ √(1+81 x²) dx

(A) Which trig substitution is correct for this integral? x = 9 cosθ x= 81 sinθ x=1/81 secθ x=1/9 secθ x=1/81 tanθ x=1/9 tanθ
(B) Which integral do you obtain after substituting for 'x' and simplifying?

Answer 1

To solve the integral of z = x³ √(1 + 81 x²) dx, the correct trigonometric substitution is x = 1/9 tanθ, which simplifies the integral into a form that is easier to evaluate involving powers of sec(θ).

Step-by-step explanation:

The student's question involves evaluating the integral of z=x³ √(1+81 x²) dx. To approach this integral, we look for a suitable trigonometric substitution that simplifies the radical expression √(1+81 x²). The correct substitution in this case is x=1/9 tanθ, which will turn the radical into a secant function that is easier to integrate.

After substituting and simplifying, the integral in terms of θ will involve trigonometric functions and powers of sec(θ). The powers of sec(θ) are then integrated using standard trigonometric identities and antiderivative rules.