Find the indefinite integral:

integral fraction numerator e to the power of 7 x end exponent over denominator e to the power of 7 x end exponent plus 36 end fraction d x


1 over 7 ln left parenthesis e to the power of 7 x end exponent plus 36 right parenthesis plus c


1 over 7 e to the power of 7 x end exponent left parenthesis e to the power of 7 x end exponent plus 36 right parenthesis plus c


7 e to the power of 7 x end exponent ln left parenthesis e to the power of 7 x end exponent plus 36 right parenthesis plus C


7 ln left parenthesis e to the power of 7 x end exponent plus 36 right parenthesis plus c

To find the indefinite integral 

\[
\int \frac{e^{7x}}{e^{7x} + 36} \, dx,
\]

we can use substitution. Let \( u = e^{7x} + 36 \). Then, the derivative \( du \) is given by:

\[
du = 7e^{7x} \, dx \quad \Rightarrow \quad dx = \frac{1}{7e^{7x}} \, du.
\]

Rewriting the integral in terms of \( u \):

\[
\int \frac{e^{7x}}{e^{7x} + 36} \, dx = \int \frac{e^{7x}}{u} \cdot \frac{1}{7e^{7x}} \, du = \frac{1}{7} \int \frac{1}{u} \, du.
\]

The integral of \( \frac{1}{u} \) is \( \ln |u| \):

\[
\frac{1}{7} \int \frac{1}{u} \, du = \frac{1}{7} \ln |u| + C.
\]

Substituting back \( u = e^{7x} + 36 \):

\[
\frac{1}{7} \ln |e^{7x} + 36| + C.
\]

Since \( e^{7x} + 36 \) is always positive, the absolute value can be omitted:

\[
\frac{1}{7} \ln (e^{7x} + 36) + C.
\]

So, the correct answer is:

\[
\frac{1}{7} \ln (e^{7x} + 36) + C.
\]

This matches the option:

\[
\frac{1}{7} \ln \left( e^{7x} + 36 \right) + C.
\]

Do you want details or have any questions?

Here are 5 questions you can ask next:
1. How do you solve integrals involving trigonometric functions?
2. Can you explain the method of partial fraction decomposition in integration?
3. How do you integrate functions involving natural logarithms?
4. What is the integral of a product of exponential and trigonometric functions?
5. How can we use integration to find the area under a curve?

Tip: When integrating rational functions, substitution can often simplify the problem by converting it into a standard integral form.

Learn how to find the indefinite integral of (e^(7x)) / (e^(7x) + 36) step-by-step using substitution method. This problem covers integration techniques involving exponential functions and reciprocal functions. Suitable for college-level mathematics.

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