Certainly! Let's go through the steps to find the antiderivative of [tex]\(\int 7 x^6 (x^7 + 9)^3 \, dx\)[/tex] using substitution. Given the substitution [tex]\(u = x^7 + 9\)[/tex], we proceed as follows.
1. Substitution and Differentiation:
First, let's find the differential [tex]\(du\)[/tex] by taking the derivative of [tex]\(u\)[/tex] with respect to [tex]\(x\)[/tex].
Since [tex]\(u = x^7 + 9\)[/tex],
[tex]\[
\frac{du}{dx} = \frac{d}{dx}(x^7 + 9) = 7x^6.
\][/tex]
Therefore,
[tex]\[
du = 7x^6 \, dx.
\][/tex]
This implies,
[tex]\[
dx = \frac{du}{7x^6}.
\][/tex]
2. Rewrite the Integral in Terms of [tex]\(u\)[/tex]:
Now, substitute [tex]\(u = x^7 + 9\)[/tex] into the integral. Also, use the relationship [tex]\(dx = \frac{du}{7x^6}\)[/tex].
The integrand [tex]\(7 x^6 (x^7 + 9)^3\)[/tex] can be rewritten as:
[tex]\[
7 x^6 u^3.
\][/tex]
Incorporating the expression for [tex]\(dx\)[/tex], we get:
[tex]\[
\int 7 x^6 u^3 \cdot \frac{du}{7x^6}.
\][/tex]
3. Simplify:
The [tex]\(7 x^6\)[/tex] terms cancel out, leaving:
[tex]\[
\int u^3 \, du.
\][/tex]
4. Find the Antiderivative:
Now, integrate with respect to [tex]\(u\)[/tex]:
[tex]\[
\int u^3 \, du = \frac{u^4}{4} + C.
\][/tex]
Finally, substitute back [tex]\(u = x^7 + 9\)[/tex] to express the answer in terms of [tex]\(x\)[/tex]:
[tex]\[
\frac{(x^7 + 9)^4}{4} + C.
\][/tex]
Therefore, the antiderivative of the given integral is:
[tex]\[
\boxed{\frac{(x^7 + 9)^4}{4} + C}.
\][/tex]
This captures the step-by-step solution for finding the antiderivative using the substitution method.
