∫ (3x² + 14x + 13) / (x + 4) dx

To solve the integral ∫ x + 4 3 x 2 + 14 x + 13 ​ d x , we will use polynomial long division and then integrate each term separately.
Step-by-step Solution:
Step 1: Perform Polynomial Long Division
First, we divide the polynomial 3 x 2 + 14 x + 13 by x + 4 .

Divide the leading term of the dividend, 3 x 2 , by the leading term of the divisor, x . This gives 3 x .

Multiply 3 x by the divisor x + 4 : 3 x ⋅ ( x + 4 ) = 3 x 2 + 12 x

Subtract this result from the original dividend: ( 3 x 2 + 14 x + 13 ) − ( 3 x 2 + 12 x ) = 2 x + 13

Next, divide the new leading term 2 x by x , which gives 2 .

Multiply 2 by x + 4 : 2 ⋅ ( x + 4 ) = 2 x + 8

Subtract this from the remaining polynomial: ( 2 x + 13 ) − ( 2 x + 8 ) = 5


So the result of the division is 3 x + 2 with a remainder of 5 . Therefore, the original expression can be rewritten as: x + 4 3 x 2 + 14 x + 13 ​ = 3 x + 2 + x + 4 5 ​
Step 2: Integrate Each Term
Now, integrate each part separately:

∫ ( 3 x + 2 ) d x which can be split into:

∫ 3 x d x = 2 3 ​ x 2
∫ 2 d x = 2 x


∫ x + 4 5 ​ d x = 5 ∫ x + 4 1 ​ d x = 5 ln ∣ x + 4∣


Final Integrated Result
Combine all these results: ∫ x + 4 3 x 2 + 14 x + 13 ​ d x = 2 3 ​ x 2 + 2 x + 5 ln ∣ x + 4∣ + C Where C is the constant of integration.
This solution involves dividing the polynomial to simplify the expression before integrating, making it easier to apply basic integration rules to each term separately.

Answered by IsabellaRoseDavis | 2025-07-06

To integrate ∫ x + 4 3 x 2 + 14 x + 13 ​ d x , first perform polynomial long division to rewrite the integrand. The result is 3 x + 2 + x + 4 5 ​ , which can be integrated separately, yielding 2 3 ​ x 2 + 2 x + 5 ln ∣ x + 4∣ + C .
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Answered by IsabellaRoseDavis | 2025-08-14