Questions: Find the general indefinite integral. (Use C for the constant [ int(4 x^2+6+frac4x^2+1) d x ]

$Find the general indefinite integral. (Use C for the constant [ int(4 x^2+6+frac4x^2+1) d x ]$

Transcript text: Find the general indefinite integral. (Use $C$ for the constant \[ \int\left(4 x^{2}+6+\frac{4}{x^{2}+1}\right) d x \]

Solution

Solution Steps

Step 1: Integrate $4x^2$

To find the integral of $4x^2$, we apply the power rule of integration: \[ \int 4x^2 \, dx = \frac{4}{3}x^3 \]

Step 2: Integrate the Constant $6$

The integral of a constant is simply the constant multiplied by the variable of integration: \[ \int 6 \, dx = 6x \]

Step 3: Integrate $\frac{4}{x^2 + 1}$

The integral of $\frac{4}{x^2 + 1}$ is a standard integral that results in: \[ \int \frac{4}{x^2 + 1} \, dx = 4 \tan^{-1}(x) \]

Step 4: Combine the Results

Combining all the results from the previous steps, we have: \[ \int \left(4x^2 + 6 + \frac{4}{x^2 + 1}\right) dx = \frac{4}{3}x^3 + 6x + 4 \tan^{-1}(x) + C \]

Final Answer

$\boxed{\frac{4}{3}x^3 + 6x + 4 \tan^{-1}(x) + C}$

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