To find the integral

$\int \frac{5}{1 + 9x^2} \, dx$

we can use a substitution to simplify it. The integral resembles the form of the standard integral:

$\int \frac{1}{a^2 + u^2} \, du = \frac{1}{a} \arctan \left( \frac{u}{a} \right) + C$

Solution

1. Rewrite the integral: $\int \frac{5}{1 + (3x)^2} \, dx$

2. Let $u = 3x$, then $du = 3 \, dx$ or $dx = \frac{du}{3}$.

3. Substitute $u$ and $dx$ into the integral: $\int \frac{5}{1 + u^2} \cdot \frac{du}{3} = \frac{5}{3} \int \frac{1}{1 + u^2} \, du$

4. Now we can integrate: $\frac{5}{3} \arctan(u) + C$

5. Substitute back $u = 3x$: $\frac{5}{3} \arctan(3x) + C$

Final Answer

$\int \frac{5}{1 + 9x^2} \, dx = \frac{5}{3} \arctan(3x) + C$

Would you like a further breakdown of this solution, or do you have any questions?

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Here are some related questions:

1. What is the integral of $\frac{1}{1+9x^2} \, dx$?
2. How do you integrate $\int \frac{5}{1+4x^2} \, dx$ using a similar method?
3. How does the substitution method work in integral calculus?
4. How does the arctan function relate to integrals of the form $\int \frac{1}{a^2 + u^2} \, du$?
5. How do you find integrals involving $\sqrt{a^2 + u^2}$?

Tip: Substitution is useful when you recognize a pattern involving a basic function and its derivative in the integrand.