# How would I use the Comparison Test in calculus to solve the integral #(cos(4x) +3) / (3x^3 + 1)# from 1 to infinity?

I'm not sure what you mean by "solve".

By comparison, we can see that the integral converges.

graph{(y-100((cos (4x)+3)/(3x^3+1)))(y-((400)/(3x^3+1)))=0 [0.33, 11.433, -0.75, 4.8]} (Both multiplied by 100 for clarity of relationship.)

So

graph{(y-100((cos (4x)+3)/(3x^3+1)))(y-((400)/(x^3)))=0 [0.2, 17.985, -1.124, 7.766]}

(Again multiplied by 100 for clarity.)

So

By signing up, you agree to our Terms of Service and Privacy Policy

To use the Comparison Test to determine the convergence of the integral (\int_{1}^{\infty} \frac{\cos(4x) + 3}{3x^3 + 1} , dx), we need to compare it with a known integral whose convergence or divergence is known.

First, note that (\frac{\cos(4x) + 3}{3x^3 + 1} \leq \frac{4}{3x^3}) for (x \geq 1) (since (\cos(4x) + 3 \leq 4) and (3x^3 + 1 \geq 3x^3)).

Now, consider the integral (\int_{1}^{\infty} \frac{4}{3x^3} , dx). This integral is a p-series with (p = 3), which converges since (p > 1).

Since (\frac{\cos(4x) + 3}{3x^3 + 1} \leq \frac{4}{3x^3}) for (x \geq 1) and the integral (\int_{1}^{\infty} \frac{4}{3x^3} , dx) converges, by the Comparison Test, the integral (\int_{1}^{\infty} \frac{\cos(4x) + 3}{3x^3 + 1} , dx) also converges.

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Using the definition of convergence, how do you prove that the sequence #{2^ -n}# converges from n=1 to infinity?
- How do you find #lim (sqrt(x+1)-1)/(sqrt(x+4)-2)# as #x->0# using l'Hospital's Rule or otherwise?
- Is the series indicated absolutely convergent, conditionally convergent, or divergent? #rarr\4-1+1/4-1/16+1/64...#
- How do you use the limit comparison test to determine if #Sigma (2n^2-1)/(3n^5+2n+1)# from #[1,oo)# is convergent or divergent?
- How do you apply the ratio test to determine if #Sigma (n!)/n^n# from #n=[1,oo)# is convergent to divergent?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7