Question

a) Use Direct Comparison Test to test the improper integral for convergence: $int_{1}^{infty} frac{sin^{2} x}{1 + x^{3}} dx$ b) Use Limit Comparison Test to test the improper integral for convergence: $int_{1}^{infty} frac{x}{x^{2} + sqrt{x} + 1} dx$

          a) Use Direct Comparison Test to test the improper integral for
convergence:
$int_{1}^{infty} frac{sin^{2} x}{1 + x^{3}} dx$
b) Use Limit Comparison Test to test the improper integral for
convergence:
$int_{1}^{infty} frac{x}{x^{2} + sqrt{x} + 1} dx$

$a) Use Direct Comparison Test to test the improper integral for convergence: int1^infty fracsin^2 x1 + x^3 dx b) Use Limit Comparison Test to test the improper integral for convergence: int1^infty fracxx^2 + sqrtx + 1 dx$

Added by Jennifer V.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

06/22/2023

Step 1

In this case, we have the function f(x) = sin(x) / (1+x). We know that 0 <= sin(x) <= 1 for all x, so we can say that 0 <= f(x) <= 1/(1+x). Now, let's consider the integral of the function g(x) = 1/(1+x) from 0 to infinity: ∫(1/(1+x)) dx from 0 to infinity We Show more…

Show all steps

Thanks for your feedback!

a) Use Direct Comparison Test to test the improper integral for convergence: ∫₁∞ (sin² x) / (1 + x³) dx b) Use Limit Comparison Test to test the improper integral for convergence: ∫₁∞ x / (x² + √xe + 1) dx