Sure, let's solve the given integral using substitution.
We are given the integral:
[tex]\[ \int 4(4x - 2)^3 \, dx \][/tex]
Step 1: Choose a substitution. Let [tex]\( u = 4x - 2 \)[/tex].
Step 2: Compute the differential [tex]\( du \)[/tex].
Since [tex]\( u = 4x - 2 \)[/tex],
[tex]\[
\frac{du}{dx} = 4 \implies du = 4 \, dx \implies dx = \frac{du}{4}
\][/tex]
Step 3: Rewrite the integral in terms of [tex]\( u \)[/tex]. Substitute [tex]\( u = 4x - 2 \)[/tex] and [tex]\( dx = \frac{du}{4} \)[/tex].
Thus, the integral becomes:
[tex]\[
\int 4(4x - 2)^3 \, dx = \int 4u^3 \cdot \frac{du}{4}
\][/tex]
Step 4: Simplify the integral. Notice that the [tex]\( 4 \)[/tex] and [tex]\( \frac{1}{4} \)[/tex] cancel out.
[tex]\[
\int u^3 \, du
\][/tex]
Step 5: Integrate [tex]\( u^3 \)[/tex].
The integral of [tex]\( u^3 \)[/tex] is:
[tex]\[
\int u^3 \, du = \frac{u^4}{4} + C
\][/tex]
Step 6: Substitute back in terms of [tex]\( x \)[/tex]. Recall that [tex]\( u = 4x - 2 \)[/tex].
So,
[tex]\[
\frac{u^4}{4} + C = \frac{(4x - 2)^4}{4} + C
\][/tex]
Therefore, the indefinite integral is:
[tex]\[
\int 4(4x - 2)^3 \, dx = \frac{(4x - 2)^4}{4} + C
\][/tex]
