Question

1. $int \frac{3}{x^{2}+14x+49}dx$

Explanation:

Step1: Factor the denominator

Notice that $x^2+14x+49=(x+7)^2$, so the integral becomes:
$$\int \frac{3}{(x+7)^2} dx$$

Step2: Use substitution

Let $u = x+7$, then $du = dx$. Substitute into the integral:
$$3\int u^{-2} du$$

Step3: Apply power rule for integration

The power rule is $\int u^n du = \frac{u^{n+1}}{n+1} + C$ for $n
eq -1$:
$$3\cdot \frac{u^{-2+1}}{-2+1} + C = 3\cdot \frac{u^{-1}}{-1} + C = -\frac{3}{u} + C$$

Step4: Substitute back $u=x+7$

Replace $u$ with the original expression:
$$-\frac{3}{x+7} + C$$

Answer:

$-\frac{3}{x+7} + C$