To evaluate the integral [tex]\(\int \frac{5 x}{3 x^2 + 7} \, dx\)[/tex], you can follow the substitution method. Here is a step-by-step solution:
1. Identify the substitution:
Let's choose [tex]\( u = 3x^2 + 7 \)[/tex]. This is because the derivative of [tex]\( 3x^2 + 7 \)[/tex] will help simplify the integrand when substituted in.
2. Differentiate [tex]\( u \)[/tex]:
[tex]\[
\frac{du}{dx} = 6x
\Rightarrow du = 6x \, dx
\][/tex]
3. Rewrite [tex]\( x \, dx \)[/tex] in terms of [tex]\( du \)[/tex]:
Since [tex]\( du = 6x \, dx \)[/tex], we can solve for [tex]\( x \, dx \)[/tex]:
[tex]\[
x \, dx = \frac{1}{6} \, du
\][/tex]
4. Substitute [tex]\( u \)[/tex] and [tex]\( du \)[/tex] into the integral:
The integral now becomes:
[tex]\[
\int \frac{5 x}{3 x^2 + 7} \, dx = \int \frac{5}{u} \cdot \frac{1}{6} \, du
\][/tex]
Simplify the constant coefficients:
[tex]\[
\int \frac{5}{u} \cdot \frac{1}{6} \, du = \frac{5}{6} \int \frac{1}{u} \, du
\][/tex]
5. Integrate with respect to [tex]\( u \)[/tex]:
The integral of [tex]\(\frac{1}{u}\)[/tex] with respect to [tex]\(u\)[/tex] is [tex]\( \ln |u| \)[/tex]:
[tex]\[
\frac{5}{6} \int \frac{1}{u} \, du = \frac{5}{6} \ln |u| + C
\][/tex]
6. Substitute back [tex]\( u = 3x^2 + 7 \)[/tex]:
Recall that [tex]\( u = 3x^2 + 7 \)[/tex], so:
[tex]\[
\frac{5}{6} \ln |u| + C = \frac{5}{6} \ln |3x^2 + 7| + C
\][/tex]
For the given options, this matches:
(C) [tex]\(\frac{5}{6} \ln \left(3 x^2 + 7\right)+c\)[/tex]
Thus, the correct answer is:
[tex]\(\boxed{\frac{5}{6} \ln \left(3 x^2 + 7\right) + c}\)[/tex]
