Question

find the indefinite integral and check the result by differentiating.

∫e¹²ˣ(12)dx

∫e¹²ˣ(12)dx=□

Explanation:

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

The integral formula is \(\int e^{ax}dx=\frac{1}{a}e^{ax}+C\) (\(a
eq0\)). 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:

\(e^{12x}+C\)