find the indefinite integral and check the result by differentiating.\n\n∫e¹²ˣ(12)dx\n\n∫e¹²ˣ(12)dx=□

Explanation:

Step1: Use the integral formula for (e^{ax})

The integral formula is (\int e^{ax}dx=\frac{1}{a}e^{ax}+C) ((a\neq0)). Here (a = 12), so (\int e^{12x}(12)dx). Let (u = 12x), then (du=12dx). By the substitution rule (\int e^{u}du=e^{u}+C). Substituting back (u = 12x), we get (\int e^{12x}(12)dx=e^{12x}+C).

Step2: Check by differentiation

Differentiate (y = e^{12x}+C) using the chain rule. If (y = e^{12x}+C), then (y^\prime=\frac{d}{dx}(e^{12x})+\frac{d}{dx}(C)). By the chain rule (\frac{d}{dx}(e^{12x})=e^{12x}\cdot\frac{d}{dx}(12x)). Since (\frac{d}{dx}(12x)=12) and (\frac{d}{dx}(C) = 0) ((C) is a constant), (y^\prime=e^{12x}\cdot12).

Answer:

(e^{12x}+C)