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int(x^(3)+4x^(2)-x+3)/(x)*dx=

22. ∫x3+4x2−x+3x⋅dx= \int \frac{x^{3}+4 x^{2}-x+3}{x} \cdot d x=

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Evaluate definite integrals using the power ruleright chevron icon

Full solution

Q. 22. ∫x3+4x2−x+3x⋅dx= \int \frac{x^{3}+4 x^{2}-x+3}{x} \cdot d x=
  1. Divide by xx: Divide each term in the numerator by xx.x3+4x2−x+3x=x2+4x−1+3x\frac{x^3 + 4x^2 - x + 3}{x} = x^2 + 4x - 1 + \frac{3}{x}
  2. Set up integral: Set up the integral of each term. ∫(x2+4x−1+3x) dx\int(x^2 + 4x - 1 + \frac{3}{x}) \, dx
  3. Integrate term by term: Integrate term by term.\newline∫x2 dx+∫4x dx−∫1 dx+∫3x dx\int x^2 \, dx + \int 4x \, dx - \int 1 \, dx + \int \frac{3}{x} \, dx\newline= 13x3+42x2−x+3ln⁡∣x∣\frac{1}{3}x^3 + \frac{4}{2}x^2 - x + 3\ln|x|
  4. Combine constants: Combine the constants in the integration. (13)x3+2x2−x+3ln⁡∣x∣(\frac{1}{3})x^3 + 2x^2 - x + 3\ln|x|
  5. Write final answer: Write the final answer.\newlineFinal Answer: (13)x3+2x2−x+3ln⁡∣x∣+C(\frac{1}{3})x^3 + 2x^2 - x + 3\ln|x| + C

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