We need to evaluate the indefinite integral of 5x plus 3 divided by x squared plus 3x plus 2. This is a rational function where the degree of the numerator is less than the degree of the denominator, making it suitable for partial fraction decomposition. We'll follow three key steps: first, factor the denominator; second, decompose into partial fractions; and third, integrate each term separately.
Now we need to factor the quadratic denominator x squared plus 3x plus 2. We look for two numbers that multiply to 2 and add to 3. These numbers are 1 and 2, since 1 times 2 equals 2, and 1 plus 2 equals 3. Therefore, x squared plus 3x plus 2 factors as x plus 1 times x plus 2. We can verify this by expanding back to get x squared plus 3x plus 2. Our integral now becomes the integral of 5x plus 3 divided by x plus 1 times x plus 2.
Now we set up the partial fraction decomposition. We write 5x plus 3 divided by x plus 1 times x plus 2 equals A over x plus 1 plus B over x plus 2. Multiplying both sides by the denominator gives us 5x plus 3 equals A times x plus 2 plus B times x plus 1. Using the substitution method, when x equals negative 1, we get negative 2 equals A, so A equals negative 2. When x equals negative 2, we get negative 7 equals negative B, so B equals 7. Therefore, our partial fraction decomposition is negative 2 over x plus 1 plus 7 over x plus 2. We can verify this by combining the fractions back to get 5x plus 3 over x plus 1 times x plus 2.
Now we integrate each partial fraction term separately. We rewrite the integral as the integral of negative 2 over x plus 1 plus 7 over x plus 2. We can split this into two separate integrals: negative 2 times the integral of 1 over x plus 1, plus 7 times the integral of 1 over x plus 2. We apply the basic integration rule that the integral of 1 over x plus a equals the natural logarithm of the absolute value of x plus a plus C. For the first term, negative 2 times the integral of 1 over x plus 1 gives us negative 2 natural log of absolute value x plus 1. For the second term, 7 times the integral of 1 over x plus 2 gives us 7 natural log of absolute value x plus 2. Combining these results, we get negative 2 natural log of absolute value x plus 1 plus 7 natural log of absolute value x plus 2 plus C.
Here is our final answer: the integral of 5x plus 3 divided by x squared plus 3x plus 2 equals negative 2 natural log of absolute value x plus 1 plus 7 natural log of absolute value x plus 2 plus C. Let's verify this result by differentiating our answer. The derivative of negative 2 natural log of absolute value x plus 1 plus 7 natural log of absolute value x plus 2 plus C equals negative 2 times 1 over x plus 1 plus 7 times 1 over x plus 2. This simplifies to negative 2 over x plus 1 plus 7 over x plus 2. Combining these fractions gives us negative 2 times x plus 2 plus 7 times x plus 1, all over x plus 1 times x plus 2. Expanding the numerator: negative 2x minus 4 plus 7x plus 7 equals 5x plus 3. So we get 5x plus 3 over x plus 1 times x plus 2, which matches our original integrand. This confirms our solution is correct. The key technique we used was partial fraction decomposition for rational functions.