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
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: $$\int \frac{2x^2 + 7x - 3}{x-2} \, dx = x^2 + 11x + 19 \ln{|x-2|} + C$$
$$\int \frac{x^4 + x + 4}{x^2 + 2} \, dx = \frac{x^3}{3} - 2x + \frac{5}{2} \ln|x^2 + 2| + 4 \sqrt{2} \arctan\left(\frac{x}{\sqrt{2}}\right) + C$$