Integration worked out questions

$$3 \left( \frac{x^3}{3} \right) - 4 \left( \frac{x^2}{2} \right) + 5x + C$$ $$x^3 - 2x^2 + 5x + C$$

The final answer is $\boxed{x^3 - 2x^2 + 5x + C}$.

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Remember $\int x^{-1} \,dx = \ln|x| + C$.

$$2 \int x^{-1} \,dx + \int x^{1/2} \,dx$$ $$2 \ln|x| + \frac{x^{1/2+1}}{1/2+1} + C$$

$$2 \ln|x| + \frac{x^{3/2}}{3/2} + C$$ $$2 \ln|x| + \frac{2}{3}x^{3/2} + C$$

The final answer is $\boxed{2 \ln|x| + \frac{2}{3}x^{3/2} + C}$.

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Then, find the differential $du$.

$$\frac{du}{dx} = 2 \implies du = 2 \,dx \implies dx = \frac{1}{2} \,du$$

$$\int u^3 \left( \frac{1}{2} \,du \right)$$ $$\frac{1}{2} \int u^3 \,du$$

$$\frac{1}{2} \left( \frac{u^{3+1}}{3+1} \right) + C$$ $$\frac{1}{2} \left( \frac{u^4}{4} \right) + C$$ $$\frac{u^4}{8} + C$$

$$\frac{(2x+1)^4}{8} + C$$

The final answer is $\boxed{\frac{(2x+1)^4}{8} + C}$.