Questions: Calcule ∫(4x^3 - 2/x^2 - 1) dx

Transcript text: Calcule $\int\left(4 x^{3}-\frac{2}{x^{2}}-1\right) d x$

Solution

Solution Steps

To solve the integral $\int\left(4 x^{3}-\frac{2}{x^{2}}-1\right) dx$, we can integrate each term separately. The integral of $4x^3$ is $\frac{4x^4}{4}$, the integral of $-\frac{2}{x^2}$ is $2x^{-1}$, and the integral of $-1$ is $-x$. Combine these results and add the constant of integration $C$.

Step 1: Integrate Each Term Separately

Para resolver a integral $\int\left(4 x^{3}-\frac{2}{x^{2}}-1\right) dx$, integramos cada termo separadamente.

Step 2: Integral of $4x^3$

A integral de $4x^3$ é: \[ \int 4x^3 \, dx = \frac{4x^4}{4} = x^4 \]

Step 3: Integral of $-\frac{2}{x^2}$

A integral de $-\frac{2}{x^2}$ é: \[ \int -\frac{2}{x^2} \, dx = -2 \int x^{-2} \, dx = -2 \left( \frac{x^{-1}}{-1} \right) = \frac{2}{x} \]

Step 4: Integral of $-1$

A integral de $-1$ é: \[ \int -1 \, dx = -x \]

Step 5: Combine the Results

Combinando os resultados das integrais, temos: \[ \int\left(4 x^{3}-\frac{2}{x^{2}}-1\right) dx = x^4 - x + \frac{2}{x} + C \]