Math / Calculus

Question $\int \frac{x^{4}+7 x^{3}+12 x^{2}+2}{x^{2}+7 x+12} d x$

Studdy Solution

STEP 1

1. The integral is a rational function, which is a polynomial divided by another polynomial.
2. The degree of the numerator is higher than the degree of the denominator, so polynomial long division will be required.
3. After division, the integral will be split into simpler parts that can be integrated separately.

STEP 2

1. Perform polynomial long division to simplify the integrand.
2. Integrate the resulting polynomial.
3. Integrate any remaining rational function.

STEP 3

Perform polynomial long division on the integrand $\frac{x^{4}+7x^{3}+12x^{2}+2}{x^{2}+7x+12}$ .
1. Divide the leading term of the numerator $x^4$ by the leading term of the denominator $x^2$ to get $x^2$ .
2. Multiply the entire denominator $x^2 + 7x + 12$ by $x^2$ and subtract from the original numerator.
3. Repeat the process with the new polynomial obtained after subtraction.

STEP 4

Continue the polynomial long division until the degree of the remainder is less than the degree of the denominator:
1. After the first subtraction, the new numerator becomes $0x^3 + 0x^2 - 12x + 2$ .
2. Divide $-12x$ by $x^2$ , which gives $-12/x$ , but since the degree is less, stop here.
The division results in: $x^2 + 0x + 0 + \frac{-12x + 2}{x^2 + 7x + 12}$

STEP 5

Integrate the polynomial part $x^2$ :
$\int x^2 \, dx = \frac{x^3}{3} + C_1$

STEP 6

Integrate the remaining rational function $\int \frac{-12x + 2}{x^2 + 7x + 12} \, dx$ .
1. Factor the denominator if possible: $x^2 + 7x + 12 = (x+3)(x+4)$ .
2. Use partial fraction decomposition to express $\frac{-12x + 2}{(x+3)(x+4)}$ as a sum of simpler fractions.
3. Integrate each term separately.

STEP 7

Perform partial fraction decomposition:
1. Express $\frac{-12x + 2}{(x+3)(x+4)}$ as $\frac{A}{x+3} + \frac{B}{x+4}$ .
2. Solve for $A$ and $B$ by equating coefficients.

STEP 8

Solve for $A$ and $B$ :
1. Multiply through by the denominator to clear fractions: $-12x + 2 = A(x+4) + B(x+3)$ .
2. Expand and collect like terms: $-12x + 2 = Ax + 4A + Bx + 3B$ .
3. Equate coefficients: - For $x$ : $-12 = A + B$ - For the constant: $2 = 4A + 3B$
Solve the system of equations to find $A$ and $B$ .

STEP 9

Solve the system of equations:
1. From $-12 = A + B$ , express $B = -12 - A$ .
2. Substitute into $2 = 4A + 3B$ : $2 = 4A + 3(-12 - A)$ $2 = 4A - 36 - 3A$ $2 = A - 36$ $A = 38$
3. Substitute $A = 38$ back to find $B$ : $B = -12 - 38 = -50$

STEP 10

Integrate each term from the partial fraction decomposition:
1. $\int \frac{38}{x+3} \, dx = 38 \ln|x+3| + C_2$
2. $\int \frac{-50}{x+4} \, dx = -50 \ln|x+4| + C_3$

STEP 11

Combine all parts of the integral:
The integral is: $\frac{x^3}{3} + 38 \ln|x+3| - 50 \ln|x+4| + C$
where $C = C_1 + C_2 + C_3$ is the constant of integration.
The solution to the integral is:
$\boxed{\frac{x^3}{3} + 38 \ln|x+3| - 50 \ln|x+4| + C}$

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