Math / Calculus

QuestionEvaluate: $\int \frac{2 x^{2}+7 x-3}{x-2} d x$ $\int \frac{x^{4}+x+4}{x^{2}+2} d x$

Studdy Solution

STEP 1

1. The given integrals involve rational functions.
2. The integrals can be solved by breaking the rational functions into simpler parts using polynomial long division and partial fraction decomposition, if necessary.
3. Standard integration techniques will be applied after simplification.

STEP 2

1. Simplify the integrand $\frac{2x^2 + 7x - 3}{x-2}$ using polynomial long division, if needed.
2. Integrate the simplified form of $\frac{2x^2 + 7x - 3}{x-2}$ .
3. Simplify the integrand $\frac{x^4 + x + 4}{x^2 + 2}$ using polynomial long division, if needed.
4. Integrate the simplified form of $\frac{x^4 + x + 4}{x^2 + 2}$ .

STEP 3

Perform polynomial long division on $\frac{2x^2 + 7x - 3}{x-2}$ .
$\begin{array}{r|rr} x - 2 & 2x^2 + 7x - 3 \\ \hline & 2x + 11 \\ \end{array}$
$2x + 11$
$2x^2 - 4x$
$11x - 3 + 4x$
$11x - 3 + 4x -3$

STEP 4

Express the result of the division as the sum of the quotient and the remainder.
$\frac{2x^2 + 7x - 3}{x-2} = 2x + 11 + \frac{19}{x-2}$

STEP 5

Integrate the expression obtained.
$\int \left(2x + 11 + \frac{19}{x-2}\right) dx$

STEP 6

Separate the integral into simpler parts and integrate each part individually.
$\int 2x \, dx + \int 11 \, dx + \int \frac{19}{x-2} \, dx$
$= 2 \int x \, dx + 11 \int dx + 19 \int \frac{1}{x-2} \, dx$

STEP 7

Solve the integrals.
$\int 2x \, dx = x^2$
$\int 11 \, dx = 11x$
$\int \frac{19}{x-2} \, dx = 19 \ln{|x-2|}$
Combining these, we get:
$\int \frac{2x^2 + 7x - 3}{x-2} \, dx = x^2 + 11x + 19 \ln{|x-2|} + C$
where $C$ is the constant of integration.

STEP 8

Perform polynomial long division on $\frac{x^4 + x + 4}{x^2 + 2}$ .
$\begin{array}{r|rr} x^2 + 2 & x^4 + x + 4 \\ \hline & x^2 - 2 + \frac{5x + 8}{x^2 + 2} \\ \end{array}$
$x^2 - 2$
$x^4 + 2 x^2 - 2x^2$
$-x^2 + x + 4$
$-x^2 - 2 + 5x + 8$

STEP 9

Express the result of the division as the sum of the quotient and the remainder.
$\frac{x^4 + x + 4}{x^2 + 2} = x^2 - 2 + \frac{5x + 8}{x^2 + 2}$

STEP 10

Integrate the expression obtained.
$\int \left(x^2 - 2 + \frac{5x + 8}{x^2 + 2}\right) dx$

STEP 11

Separate the integral into simpler parts and integrate each part individually.
$\int x^2 \, dx - \int 2 \, dx + \int \frac{5x}{x^2 + 2} \, dx + \int \frac{8}{x^2 + 2} \, dx$
$= \int x^2 \, dx - 2 \int dx + 5 \int \frac{x}{x^2 + 2} \, dx + 8 \int \frac{1}{x^2 + 2} \, dx$

STEP 12

Solve the integrals.
$\int x^2 \, dx = \frac{x^3}{3}$
$2 \int dx = 2x$
For the third term, use the substitution $u = x^2 + 2$ , $du = 2x \, dx$ :
$5 \int \frac{x}{x^2 + 2} \, dx = \frac{5}{2} \int \frac{1}{u} \, du = \frac{5}{2} \ln|u| = \frac{5}{2} \ln|x^2 + 2|$
For the fourth term, notice that $8 \int \frac{1}{x^2 + 2} \, dx$ can be solved by recognizing it as a standard integral form:
$8 \int \frac{1}{x^2 + 2} \, dx = 8 \cdot \frac{1}{\sqrt{2}} \arctan\left(\frac{x}{\sqrt{2}}\right) = 4 \sqrt{2} \arctan\left(\frac{x}{\sqrt{2}}\right)$
Combining these, we get:
$\int \frac{x^4 + x + 4}{x^2 + 2} \, dx = \frac{x^3}{3} - 2x + \frac{5}{2} \ln|x^2 + 2| + 4 \sqrt{2} \arctan\left(\frac{x}{\sqrt{2}}\right) + C$
where $C$ is the constant of integration.
Solution: $\int \frac{2x^2 + 7x - 3}{x-2} \, dx = x^2 + 11x + 19 \ln{|x-2|} + C$
$\int \frac{x^4 + x + 4}{x^2 + 2} \, dx = \frac{x^3}{3} - 2x + \frac{5}{2} \ln|x^2 + 2| + 4 \sqrt{2} \arctan\left(\frac{x}{\sqrt{2}}\right) + C$

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