ts) Integrate int(x^(3)+x)/(x^(2)+2)dx.

Question

ts) Integrate  int(x^(3)+x)/(x^(2)+2)dx.

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Simplify Integrand: Simplify the integrand by dividing each term in the numerator by each term in the denominator. We have the integral: int((x^3 + x) / (x^2 + 2))dx We can split this into two separate integrals: int(x^3 / (x^2 + 2))dx + int(x / (x^2 + 2))dxSplit into Two Integrals: Simplify the first integral by recognizing that the numerator is the derivative of the denominator. For the first integral, we have: int(x^3 / (x^2 + 2))dx Let u = x^2 + 2, then du = 2x dx, which means x dx = du/2. Substitute u and du into the integral: 1/2 * int(u - 2) / u duSubstitute and Integrate: Simplify the second integral by recognizing it as a standard form. For the second integral, we have: int(x / (x^2 + 2))dx This is a standard form that can be integrated directly. Let's integrate it: 1/2 * ln|x^2 + 2| + CIntegrate First Integral: Integrate the first integral after substitution. We have: 1/2 * int(u - 2) / u du Split the integral: 1/2 * (int(1)du - 2 * int(1/u)du) Integrate term by term: 1/2 * (u - 2ln|u|) + CSubstitute Back: Substitute back the original variable x into the first integral. We have: 1/2 * (u - 2ln|u|) + C Substitute back u = x^2 + 2: 1/2 * ((x^2 + 2) - 2ln|x^2 + 2|) + CCombine Final Answer: Combine the results from Step 3 and Step 5 to get the final answer. We have: 1/2 * ((x^2 + 2) - 2ln|x^2 + 2|) + 1/2 * ln|x^2 + 2| + C Simplify the expression: 1/2 * x^2 + 1 - ln|x^2 + 2| + C

Integrate $\int\frac{x^{3}+x}{x^{2}+2}\,dx$ .

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Integrate $\int\frac{x^{3}+x}{x^{2}+2}\,dx$ .