Certainly! Let's consider the integral [tex]\(\int 7^{2x} \, dx\)[/tex].
To solve this integral, we can use a substitution method and properties of logarithms. Let's follow these steps:
1. Substitute:
Let [tex]\(u = 2x\)[/tex]. Then, [tex]\(du = 2dx\)[/tex] or [tex]\(dx = \frac{1}{2}du\)[/tex].
2. Rewrite the integral:
Substitute [tex]\(u\)[/tex] and [tex]\(dx\)[/tex] into the integral:
[tex]\[
\int 7^{2x} \, dx = \int 7^u \cdot \frac{1}{2} \, du
\][/tex]
3. Rewrite constants:
Simplify the expression by bringing the constant outside the integral:
[tex]\[
\int 7^u \cdot \frac{1}{2} \, du = \frac{1}{2} \int 7^u \, du
\][/tex]
4. Integrate:
The integral of [tex]\(7^u\)[/tex] with respect to [tex]\(u\)[/tex] is:
[tex]\[
\int 7^u \, du = \frac{7^u}{\log(7)} + C
\][/tex]
Where [tex]\(C\)[/tex] is the constant of integration, and [tex]\(\log(7)\)[/tex] represents the natural logarithm of 7.
5. Include the constant:
Now, include the constant back into our integral:
[tex]\[
\frac{1}{2} \int 7^u \, du = \frac{1}{2} \left( \frac{7^u}{\log(7)} + C \right) = \frac{7^u}{2 \log(7)} + \frac{C}{2}
\][/tex]
6. Back-substitute [tex]\(u = 2x\)[/tex]:
Finally, replace [tex]\(u\)[/tex] with the original variable. Since [tex]\(u = 2x\)[/tex], we get:
[tex]\[
\frac{7^u}{2 \log(7)} + \frac{C}{2} = \frac{7^{2x}}{2 \log(7)} + C
\][/tex]
(Note that [tex]\(\frac{C}{2}\)[/tex] is just another constant, so we can denote it as [tex]\(C\)[/tex] for simplicity.)
Therefore, the evaluated integral is:
[tex]\[
\int 7^{2x} \, dx = \frac{7^{2x}}{2 \log(7)} + C
\][/tex]
