To solve the integral [tex]\(\int 2bx^3 \, dx\)[/tex], we follow a series of steps to find the antiderivative.
1. Identify the integrand:
The integrand is [tex]\(2bx^3\)[/tex], where [tex]\(b\)[/tex] is a constant and [tex]\(x\)[/tex] is the variable of integration.
2. Apply the power rule for integration:
The power rule for integration states that [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex], where [tex]\(C\)[/tex] is the constant of integration, and [tex]\(n \neq -1\)[/tex].
3. Factor out the constant [tex]\(b\)[/tex]:
Since [tex]\(b\)[/tex] is a constant, we can factor it out of the integral:
[tex]\[
\int 2bx^3 \, dx = 2b \int x^3 \, dx
\][/tex]
4. Integrate the remaining function [tex]\(x^3\)[/tex]:
Using the power rule for integration:
[tex]\[
\int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4}
\][/tex]
5. Multiply by the constant [tex]\(2b\)[/tex]:
Now we multiply the result by [tex]\(2b\)[/tex]:
[tex]\[
2b \cdot \frac{x^4}{4} = \frac{2b x^4}{4} = \frac{b x^4}{2}
\][/tex]
6. Include the constant of integration [tex]\(C\)[/tex]:
The final antiderivative includes an arbitrary constant [tex]\(C\)[/tex]:
[tex]\[
\int 2bx^3 \, dx = \frac{b x^4}{2} + C
\][/tex]
So, the result of the integral [tex]\(\int 2bx^3 \, dx\)[/tex] is:
[tex]\[
\frac{b x^4}{2} + C
\][/tex]
