Sure, let's find the antiderivative of [tex]\(\int 7x^6 (x^7 + 9)^3 \, dx\)[/tex] using substitution.
First, we make the substitution [tex]\( u = x^7 + 9 \)[/tex].
Then, we find the differential [tex]\( du \)[/tex].
To do this, we need to take the derivative of [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
u = x^7 + 9
\][/tex]
Differentiating both sides with respect to [tex]\( x \)[/tex], we get:
[tex]\[
\frac{du}{dx} = 7x^6
\][/tex]
Multiplying both sides by [tex]\( dx \)[/tex]:
[tex]\[
du = 7x^6 \, dx
\][/tex]
This tells us that [tex]\( 7x^6 \, dx \)[/tex] is equivalent to [tex]\( du \)[/tex].
Next, we substitute [tex]\( u \)[/tex] and [tex]\( du \)[/tex] into the integral:
[tex]\[
\int 7x^6 (x^7 + 9)^3 \, dx
\][/tex]
becomes:
[tex]\[
\int (x^7 + 9)^3 \cdot 7x^6 \, dx
\][/tex]
which simplifies to:
[tex]\[
\int u^3 \, du
\][/tex]
Now we need to find the antiderivative of [tex]\( u^3 \)[/tex]:
[tex]\[
\int u^3 \, du = \frac{u^4}{4} + C
\][/tex]
Finally, we substitute back [tex]\( u = x^7 + 9 \)[/tex] into our result to get the antiderivative in terms of [tex]\( x \)[/tex]:
[tex]\[
\frac{u^4}{4} + C = \frac{(x^7 + 9)^4}{4} + C
\][/tex]
Therefore, the antiderivative of [tex]\(\int 7x^6 (x^7 + 9)^3 \, dx\)[/tex] is:
[tex]\[
\boxed{\frac{(x^7 + 9)^4}{4} + C}
\][/tex]
