Question

Int (dx)/(x^2+10x+29)

Answer 1

Ответ:

53456678885443467

Пошаговое объяснение:

Answer 2

The integral you've provided can be solved using partial fraction decomposition. The integrand is a rational function, and we want to decompose it into simpler fractions that can be integrated more easily. Here's how you can do it step by step:

Given integral: ∫(dx) / (x^2 + 10x + 29)

1. Start by factoring the denominator: x^2 + 10x + 29 = (x + 5)^2 + 4^2.

2. Notice that the denominator can be written as the sum of squares, which suggests using a trigonometric substitution.

3. Let's make a substitution: x + 5 = 4 * tan(θ), which implies dx = 4 * sec^2(θ) dθ.

4. Substitute the values into the integral: ∫(4 * sec^2(θ) dθ) / (16 * tan^2(θ) + 16).

5. Simplify the expression by canceling out common factors: ∫(sec^2(θ) dθ) / (tan^2(θ) + 1).

6. Use the trigonometric identity: tan^2(θ) + 1 = sec^2(θ).

7. Now, the integral becomes: ∫(sec^2(θ) dθ) / sec^2(θ), which simplifies to ∫dθ.

8. Integrate with respect to θ: ∫dθ = θ + C, where C is the constant of integration.

9. Substitute back θ = arctan((x + 5) / 4): ∫(dx) / (x^2 + 10x + 29) = arctan((x + 5) / 4) + C.

So, the integral of (dx) / (x^2 + 10x + 29) is arctan((x + 5) / 4) + C, where C is the constant of integration.