To find the indefinite integral [tex]\(\int \frac{x^4}{7 - x^5}\, dx\)[/tex], follow these detailed, step-by-step instructions:
1. Identify the integrand:
The given integral is:
[tex]\[
\int \frac{x^4}{7 - x^5}\, dx
\][/tex]
2. Consider a substitution:
Let's use the substitution [tex]\(u = x^5\)[/tex]. This choice is motivated by the [tex]\(x^5\)[/tex] term in the denominator. Under this substitution:
[tex]\[
du = 5x^4 \, dx \quad \text{or} \quad dx = \frac{du}{5x^4}
\][/tex]
3. Rewrite [tex]\(x\)[/tex] in terms of [tex]\(u\)[/tex]:
Since [tex]\(u = x^5\)[/tex], we can write [tex]\(x^4 = u^{4/5}\)[/tex].
4. Substitute into the integrand:
Substitute [tex]\(x^4\)[/tex] and [tex]\(dx\)[/tex] in the integral:
[tex]\[
\int \frac{x^4}{7 - x^5}\, dx = \int \frac{u^{4/5}}{7 - u} \cdot \frac{du}{5x^4} = \int \frac{u^{4/5}}{7 - u} \cdot \frac{du}{5u^{4/5}} = \int \frac{1}{5(7 - u)} \, du
\][/tex]
Here, [tex]\(x^4\)[/tex] and [tex]\(u^{4/5}\)[/tex] cancel out.
5. Simplify the integrand:
Factor the constant [tex]\(\frac{1}{5}\)[/tex] out of the integral:
[tex]\[
\frac{1}{5} \int \frac{1}{7 - u}\, du
\][/tex]
6. Integrate:
The integral [tex]\(\int \frac{1}{7 - u}\, du\)[/tex] can be solved using the natural logarithm function:
[tex]\[
\int \frac{1}{7 - u}\, du = -\ln |7 - u|
\][/tex]
Don't forget to include the constant of integration [tex]\(C\)[/tex]:
[tex]\[
\int \frac{1}{7 - u}\, du = -\ln |7 - u| + C
\][/tex]
7. Substitute back [tex]\(u = x^5\)[/tex]:
Replace [tex]\(u\)[/tex] with the original variable [tex]\(x\)[/tex]:
[tex]\[
-\ln |7 - x^5| + C
\][/tex]
8. Adjust the sign inside the logarithm if necessary:
Since [tex]\(\ln |a| = \ln | -a|\)[/tex], we can adjust the sign inside the logarithm for simplification:
[tex]\[
-\ln |7 - x^5| = \ln |x^5 - 7|
\][/tex]
9. Final result:
Thus, the integral evaluates to:
[tex]\[
\int \frac{x^4}{7 - x^5}\, dx = \frac{1}{5} \ln |x^5 - 7| + C
\][/tex]
Therefore, the indefinite integral is:
[tex]\[
\int \frac{x^4}{7 - x^5}\, dx = \frac{1}{5} \ln |x^5 - 7| + C
\][/tex]
