Final answer:
The integral ∫x39−x2dx can be evaluated using trigonometric substitution. The correct substitution for this integral is x=3sin(θ), which simplifies the integrand based on trigonometric identities and the geometry of right-angle triangles.
the corret opction is:a.b.c.d.e.f
Step-by-step explanation:
The integral to evaluate is ∫x39−x2dx, and the correct trigonometric substitution to use is based on the form of the integrand.
The integrand contains a term −x2, suggesting the use of a trigonometric identity that involves 1 − sin²(θ) or 1 − tan²(θ), which are equivalent to cos²(θ) and sec²(θ), respectively.
The substitution x = 3sin(θ) is appropriate because it matches the form of −x2 in the integrand, when squared, resulting in 9 - 9sin²(θ) = 9cos²(θ), which simplifies the integral.
To correctly apply this substitution, we would express the integrand in terms of θ, replace dx with d(3sin(θ)), and use trigonometric identities and the geometry of right-angle triangles, including potentially the law of sines or cosines, to integrate effectively.
Therefore, the correct answer is:a. x=3sin(θ)
The complete question is:content loaded
Evaluate the integral: ∫x39−x2dx (A) Which trig substitution is correct for this integral?
a x=3sin(θ)
b x=9sec(θ)
c x=9tan(θ)
d x=3sec(θ)
e x=9sin(θ)
f x=3tan(θ)