Connect with a community that values knowledge and expertise on IDNLearn.com. Ask your questions and get detailed, reliable answers from our community of knowledgeable experts.

Evaluate the following integral. Be sure to check by differentiating.

[tex]\[
\int e^{2x+6} \, dx
\][/tex]

[tex]\[
\int e^{2x+6} \, dx =
\][/tex]

(Type an exact answer. Use parentheses to clearly denote the argument of each function.)


Sagot :

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.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For dependable and accurate answers, visit IDNLearn.com. Thanks for visiting, and see you next time for more helpful information.