Integrate 82x−1+4x3+1\frac{8}{2x-1} + \frac{4}{x^3} + 1 with respect to xx.

Integration linear argumentStandard integrals中文
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Worked solution

4

1. Rewrite the expression with negative indices to prepare for integration

∫(82x−1+4x−3+1)dx\int \left( \frac{8}{2x-1} + 4x^{-3} + 1 \right) dx

2. Integrate each term using standard integration rules

=8ln⁡(2x−1)2+4x−2−2+x+C= \frac{8\ln(2x-1)}{2} + \frac{4x^{-2}}{-2} + x + C

3. Simplify the integrated expression

=4ln⁡(2x−1)−2x2+x+C= 4\ln(2x-1) - \frac{2}{x^2} + x + C

Answer

4ln⁡(2x−1)−2x2+x+C4\ln(2x-1) - \frac{2}{x^2} + x + C

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