Questions: Use integration by substitution to solve the integral below. Use C for the constant of integration. [ int frac6 x+21x^2+7 x+5 d x ]

$Use integration by substitution to solve the integral below. Use C for the constant of integration. [ int frac6 x+21x^2+7 x+5 d x ]$

Transcript text: Use integration by substitution to solve the integral below. Use $C$ for the constant of integration. \[ \int \frac{6 x+21}{x^{2}+7 x+5} d x \]

Solution

Solution Steps

Step 1: Choose a substitution

Let $u = x^2 + 7x + 5$.

Step 2: Find the derivative of the substitution

Then, $\frac{du}{dx} = 2x + 7$, which implies $du = (2x+7)dx$.

Step 3: Rewrite the integral in terms of _u_

Notice that $6x + 21 = 3(2x+7)$. Then we can rewrite the integral as: \[ \int \frac{6x+21}{x^{2}+7 x+5} d x = \int \frac{3(2x+7)}{x^{2}+7 x+5} d x = 3\int \frac{2x+7}{x^2+7x+5}dx = 3\int \frac{1}{u}du \]

Step 4: Solve the integral in terms of _u_

\[ 3\int \frac{1}{u} du = 3 \ln|u| + C \]

Step 5: Substitute back the original variable

Substituting $u = x^2 + 7x + 5$ back into the result, we get: \[ 3 \ln|x^2 + 7x + 5| + C \]