Break down into two integrals: Break down the integral into two separate integrals. int(4e^(3x) + 1)dx = int(4e^(3x))dx + int(1)dxIntegrate 4e^(3x): Integrate the first part int(4e^(3x))dx. To integrate 4e^(3x), we use the substitution method. Let u = 3x, which implies du/dx = 3 or dx = du/3. Therefore, the integral becomes (4/3)int(e^u)du.Perform integration of e^u: Perform the integration of e^u. The integral of e^u with respect to u is e^u. So, (4/3)int(e^u)du = (4/3)e^u + C, where C is the constant of integration.Substitute back u = 3x: Substitute back u = 3x into the integral. Substituting back, we get (4/3)e^(3x) + C.Integrate 1: Integrate the second part int(1)dx. The integral of 1 with respect to x is x. So, int(1)dx = x + C', where C' is another constant of integration.Combine results: Combine the results from Step 4 and Step 5. The combined integral is (4/3)e^(3x) + x + C'', where C'' is a new constant of integration that combines C and C'.Write final answer: Write the final answer. The indefinite integral of the function 4e^(3x) + 1 with respect to x is (4/3)e^(3x) + x + C.

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$\int\left(4 e^{3 x}+1\right) d x$

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\int\left(4 e^{3 x}+1\right) d x

Break down into two integrals: Break down the integral into two separate integrals. \int(\(4e^{ $3$ x} + $1$ )\,dx = \int( $4$ e^{ $3$ x})\,dx + \int( $1$ )\,dx
Integrate $4e^{3x}$ : Integrate the first part $\int(4e^{3x})\,dx$ . To integrate $4e^{3x}$ , we use the substitution method. Let $u = 3x$ , which implies $\frac{du}{dx} = 3$ or $dx = \frac{du}{3}$ . Therefore, the integral becomes $\left(\frac{4}{3}\right)\int(e^u)\,du$ .
Perform integration of $e^u$ : Perform the integration of $e^u$ . The integral of $e^u$ with respect to $u$ is $e^u$ . So, $(4/3)\int(e^u)\,du = (4/3)e^u + C$ , where $C$ is the constant of integration.
Substitute back $u = 3x$ : Substitute back $u = 3x$ into the integral. $\newline$ Substituting back, we get $(\frac{4}{3})e^{(3x)} + C$ .
Integrate $1$ : Integrate the second part $\int(1)\,dx$ . The integral of $1$ with respect to $x$ is $x$ . So, $\int(1)\,dx = x + C'$ , where $C'$ is another constant of integration.
Combine results: Combine the results from Step $4$ and Step $5$ . $\newline$ The combined integral is $(\frac{4}{3})e^{(3x)} + x + C''$ , where $C''$ is a new constant of integration that combines $C$ and $C'$ .
Write final answer: Write the final answer. $\newline$ The indefinite integral of the function $4e^{3x} + 1$ with respect to $x$ is $(\frac{4}{3})e^{3x} + x + C$ .