To evaluate the integral [tex]\(\int e^{2x+6} \, dx\)[/tex], we will proceed as follows:
1. Identify the integrand: The function we want to integrate is [tex]\(e^{2x+6}\)[/tex].
2. Use substitution: Let [tex]\(u = 2x + 6\)[/tex]. Then, the derivative of [tex]\(u\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
\frac{du}{dx} = 2
\][/tex]
Solving this for [tex]\(dx\)[/tex], we get:
[tex]\[
dx = \frac{du}{2}
\][/tex]
3. Rewrite the integral in terms of [tex]\(u\)[/tex]:
Substituting [tex]\(u = 2x + 6\)[/tex] and [tex]\(dx = \frac{du}{2}\)[/tex] into the integral, we get:
[tex]\[
\int e^{2x+6} \, dx = \int e^u \frac{du}{2}
\][/tex]
Simplifying, this becomes:
[tex]\[
\frac{1}{2} \int e^u \, du
\][/tex]
4. Integrate: The integral of [tex]\(e^u\)[/tex] with respect to [tex]\(u\)[/tex] is:
[tex]\[
\int e^u \, du = e^u
\][/tex]
Therefore,
[tex]\[
\frac{1}{2} \int e^u \, du = \frac{1}{2} e^u
\][/tex]
5. Substitute back: Replace [tex]\(u\)[/tex] with the original expression [tex]\(2x + 6\)[/tex]:
[tex]\[
\frac{1}{2} e^u = \frac{1}{2} e^{2x+6}
\][/tex]
6. Include the constant of integration: In indefinite integrals, we must add a constant [tex]\(C\)[/tex]:
[tex]\[
\int e^{2x+6} \, dx = \frac{1}{2} e^{2x+6} + C
\][/tex]
Therefore, the exact answer is:
[tex]\[
\int e^{2x+6} \, dx = \frac{1}{2} e^{2x+6} + C
\][/tex]
Check by Differentiating:
To ensure our solution is correct, we differentiate [tex]\(\frac{1}{2} e^{2x+6} + C\)[/tex] with respect to [tex]\(x\)[/tex]:
[tex]\[
\frac{d}{dx} \left( \frac{1}{2} e^{2x+6} + C \right)
\][/tex]
Since the derivative of a constant is zero, we only need to differentiate [tex]\(\frac{1}{2} e^{2x+6}\)[/tex]:
[tex]\[
\frac{d}{dx} \left( \frac{1}{2} e^{2x+6} \right) = \frac{1}{2} \cdot e^{2x+6} \cdot \frac{d}{dx} (2x+6)
\][/tex]
The derivative of [tex]\(2x+6\)[/tex] is 2, so:
[tex]\[
\frac{d}{dx} \left( \frac{1}{2} e^{2x+6} \right) = \frac{1}{2} \cdot e^{2x+6} \cdot 2 = e^{2x+6}
\][/tex]
Thus, our result:
[tex]\[
\int e^{2x+6} \, dx = \frac{1}{2} e^{2x+6} + C
\][/tex]
is verified as correct.
